| L(s) = 1 | + 4·2-s + 4·3-s + 8·4-s − 4·5-s + 16·6-s + 3·7-s + 18·8-s + 13·9-s − 16·10-s + 2·11-s + 32·12-s + 4·13-s + 12·14-s − 16·15-s + 34·16-s − 2·17-s + 52·18-s + 5·19-s − 32·20-s + 12·21-s + 8·22-s − 28·23-s + 72·24-s + 26·25-s + 16·26-s + 32·27-s + 24·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 4·4-s − 1.78·5-s + 6.53·6-s + 1.13·7-s + 6.36·8-s + 13/3·9-s − 5.05·10-s + 0.603·11-s + 9.23·12-s + 1.10·13-s + 3.20·14-s − 4.13·15-s + 17/2·16-s − 0.485·17-s + 12.2·18-s + 1.14·19-s − 7.15·20-s + 2.61·21-s + 1.70·22-s − 5.83·23-s + 14.6·24-s + 26/5·25-s + 3.13·26-s + 6.15·27-s + 4.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.49464103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.49464103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( ( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 - 4 T + p T^{2} + 8 T^{3} - 4 p T^{4} + 8 T^{5} - 19 T^{6} - 22 p T^{7} + 283 T^{8} - 22 p^{2} T^{9} - 19 p^{2} T^{10} + 8 p^{3} T^{11} - 4 p^{5} T^{12} + 8 p^{5} T^{13} + p^{7} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 7 | \( 1 - 3 T - 3 T^{2} + 34 T^{3} - 102 T^{4} + 181 T^{5} - 26 T^{6} - 264 p T^{7} + 8023 T^{8} - 264 p^{2} T^{9} - 26 p^{2} T^{10} + 181 p^{3} T^{11} - 102 p^{4} T^{12} + 34 p^{5} T^{13} - 3 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 2 T + p T^{2} - 58 T^{3} + 116 T^{4} + 164 T^{5} + 39 p T^{6} + 4756 T^{7} - 24153 T^{8} + 4756 p T^{9} + 39 p^{3} T^{10} + 164 p^{3} T^{11} + 116 p^{4} T^{12} - 58 p^{5} T^{13} + p^{7} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 - 4 T + p T^{2} + 88 T^{3} - 532 T^{4} + 1928 T^{5} - 109 T^{6} - 23966 T^{7} + 124003 T^{8} - 23966 p T^{9} - 109 p^{2} T^{10} + 1928 p^{3} T^{11} - 532 p^{4} T^{12} + 88 p^{5} T^{13} + p^{7} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( 1 + 2 T - 3 T^{2} - 126 T^{3} - 532 T^{4} - 84 T^{5} + 3479 T^{6} + 28082 T^{7} + 52803 T^{8} + 28082 p T^{9} + 3479 p^{2} T^{10} - 84 p^{3} T^{11} - 532 p^{4} T^{12} - 126 p^{5} T^{13} - 3 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 5 T + 29 T^{2} + 70 T^{3} - 590 T^{4} + 4435 T^{5} - 2714 T^{6} - 42900 T^{7} + 426339 T^{8} - 42900 p T^{9} - 2714 p^{2} T^{10} + 4435 p^{3} T^{11} - 590 p^{4} T^{12} + 70 p^{5} T^{13} + 29 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 14 T + 113 T^{2} + 750 T^{3} + 4121 T^{4} + 750 p T^{5} + 113 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 10 T + 11 T^{2} - 10 p T^{3} - 2239 T^{4} - 10 p^{2} T^{5} + 11 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 - 7 T + p T^{2} - 518 T^{3} + 3626 T^{4} - 15911 T^{5} + 3954 p T^{6} - 1177414 T^{7} + 5416137 T^{8} - 1177414 p T^{9} + 3954 p^{3} T^{10} - 15911 p^{3} T^{11} + 3626 p^{4} T^{12} - 518 p^{5} T^{13} + p^{7} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 + 6 T + 63 T^{2} - 22 T^{3} - 372 T^{4} - 19372 T^{5} - 833 p T^{6} - 91506 T^{7} + 3060523 T^{8} - 91506 p T^{9} - 833 p^{3} T^{10} - 19372 p^{3} T^{11} - 372 p^{4} T^{12} - 22 p^{5} T^{13} + 63 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 12 T + 17 T^{2} + 60 T^{3} + 961 T^{4} + 60 p T^{5} + 17 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 16 T + 213 T^{2} + 2088 T^{3} + 21008 T^{4} + 187488 T^{5} + 1667411 T^{6} + 13076524 T^{7} + 100516383 T^{8} + 13076524 p T^{9} + 1667411 p^{2} T^{10} + 187488 p^{3} T^{11} + 21008 p^{4} T^{12} + 2088 p^{5} T^{13} + 213 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 5 T + 69 T^{2} - 270 T^{3} - 1990 T^{4} - 37635 T^{5} - 145834 T^{6} + 485900 T^{7} + 5823099 T^{8} + 485900 p T^{9} - 145834 p^{2} T^{10} - 37635 p^{3} T^{11} - 1990 p^{4} T^{12} - 270 p^{5} T^{13} + 69 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 6 T + 6 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( 1 + 23 T + 221 T^{2} - 18 T^{3} - 21614 T^{4} - 255201 T^{5} - 803866 T^{6} + 12228416 T^{7} + 191155087 T^{8} + 12228416 p T^{9} - 803866 p^{2} T^{10} - 255201 p^{3} T^{11} - 21614 p^{4} T^{12} - 18 p^{5} T^{13} + 221 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 - 14 T + 193 T^{2} - 1162 T^{3} + 10388 T^{4} - 28532 T^{5} + 925511 T^{6} - 7337176 T^{7} + 107602183 T^{8} - 7337176 p T^{9} + 925511 p^{2} T^{10} - 28532 p^{3} T^{11} + 10388 p^{4} T^{12} - 1162 p^{5} T^{13} + 193 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 20 T + 319 T^{2} + 2800 T^{3} + 25940 T^{4} + 208760 T^{5} + 2905241 T^{6} + 31993630 T^{7} + 344475259 T^{8} + 31993630 p T^{9} + 2905241 p^{2} T^{10} + 208760 p^{3} T^{11} + 25940 p^{4} T^{12} + 2800 p^{5} T^{13} + 319 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 - 14 T + 3 T^{2} + 1578 T^{3} - 19132 T^{4} + 149148 T^{5} - 347179 T^{6} - 14598716 T^{7} + 238036263 T^{8} - 14598716 p T^{9} - 347179 p^{2} T^{10} + 149148 p^{3} T^{11} - 19132 p^{4} T^{12} + 1578 p^{5} T^{13} + 3 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 20 T + 71 T^{2} - 690 T^{3} - 7699 T^{4} - 690 p T^{5} + 71 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 27 T + 192 T^{2} + 955 T^{3} - 23289 T^{4} + 955 p T^{5} + 192 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.20789034638780961464360002948, −4.17032051684739279033390532836, −4.08682216959348296336814799820, −4.08164493817898073026091465308, −3.85167967245079494714639431891, −3.83265929505315384112028756664, −3.78748055567458989427851537287, −3.36488116422266350046269205513, −3.24925498478259712805312822736, −3.10415599124691088442693116162, −3.10166109113870133955242627124, −3.04712901743664340291326271241, −2.87612019085200685385944420081, −2.43727722299907235673867468938, −2.33551682088031588461351314441, −2.21704725252552378320276771094, −2.04350384877809987479288314283, −1.78185954637011050749855753362, −1.74497410112927627896991244262, −1.71304824515529950940436722829, −1.64082479956893397083512742149, −1.15887946422926004521983846191, −1.14012710790223709121980708784, −0.862794986317988960977645569531, −0.11607423535526923119251637674,
0.11607423535526923119251637674, 0.862794986317988960977645569531, 1.14012710790223709121980708784, 1.15887946422926004521983846191, 1.64082479956893397083512742149, 1.71304824515529950940436722829, 1.74497410112927627896991244262, 1.78185954637011050749855753362, 2.04350384877809987479288314283, 2.21704725252552378320276771094, 2.33551682088031588461351314441, 2.43727722299907235673867468938, 2.87612019085200685385944420081, 3.04712901743664340291326271241, 3.10166109113870133955242627124, 3.10415599124691088442693116162, 3.24925498478259712805312822736, 3.36488116422266350046269205513, 3.78748055567458989427851537287, 3.83265929505315384112028756664, 3.85167967245079494714639431891, 4.08164493817898073026091465308, 4.08682216959348296336814799820, 4.17032051684739279033390532836, 4.20789034638780961464360002948
Plot not available for L-functions of degree greater than 10.
