| L(s) = 1 | − 4·2-s − 4·3-s + 6·4-s + 4·5-s + 16·6-s + 6·9-s − 16·10-s − 6·11-s − 24·12-s + 12·13-s − 16·15-s − 15·16-s + 10·17-s − 24·18-s + 4·19-s + 24·20-s + 24·22-s − 4·23-s + 10·25-s − 48·26-s + 12·29-s + 64·30-s − 2·31-s + 24·32-s + 24·33-s − 40·34-s + 36·36-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 3·4-s + 1.78·5-s + 6.53·6-s + 2·9-s − 5.05·10-s − 1.80·11-s − 6.92·12-s + 3.32·13-s − 4.13·15-s − 3.75·16-s + 2.42·17-s − 5.65·18-s + 0.917·19-s + 5.36·20-s + 5.11·22-s − 0.834·23-s + 2·25-s − 9.41·26-s + 2.22·29-s + 11.6·30-s − 0.359·31-s + 4.24·32-s + 4.17·33-s − 6.85·34-s + 6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236363397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236363397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 3 | \( ( 1 + T + T^{2} )^{4} \) |
| 7 | \( 1 - 12 T^{3} + 53 T^{4} - 12 p T^{5} + p^{4} T^{8} \) |
| 23 | \( ( 1 + T + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 2 T + T^{2} + 14 T^{3} - 36 T^{4} + 14 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} + 6 p T^{3} + 23 p T^{4} + 72 T^{5} + 2042 T^{6} + 8076 T^{7} + 562 T^{8} + 8076 p T^{9} + 2042 p^{2} T^{10} + 72 p^{3} T^{11} + 23 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 6 T + 51 T^{2} - 198 T^{3} + 986 T^{4} - 198 p T^{5} + 51 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 10 T + p T^{2} + 54 T^{3} + 371 T^{4} - 2240 T^{5} - 5936 T^{6} + 1824 p T^{7} - 11426 T^{8} + 1824 p^{2} T^{9} - 5936 p^{2} T^{10} - 2240 p^{3} T^{11} + 371 p^{4} T^{12} + 54 p^{5} T^{13} + p^{7} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 4 T - 30 T^{2} + 208 T^{3} + 242 T^{4} - 4020 T^{5} + 6592 T^{6} + 35108 T^{7} - 216513 T^{8} + 35108 p T^{9} + 6592 p^{2} T^{10} - 4020 p^{3} T^{11} + 242 p^{4} T^{12} + 208 p^{5} T^{13} - 30 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 6 T + 115 T^{2} - 498 T^{3} + 5004 T^{4} - 498 p T^{5} + 115 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 2 T - 53 T^{2} - 698 T^{3} + 1013 T^{4} + 30032 T^{5} + 168050 T^{6} - 687524 T^{7} - 6889622 T^{8} - 687524 p T^{9} + 168050 p^{2} T^{10} + 30032 p^{3} T^{11} + 1013 p^{4} T^{12} - 698 p^{5} T^{13} - 53 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 90 T^{2} + 144 T^{3} + 4042 T^{4} - 9144 T^{5} - 113616 T^{6} + 187344 T^{7} + 3312495 T^{8} + 187344 p T^{9} - 113616 p^{2} T^{10} - 9144 p^{3} T^{11} + 4042 p^{4} T^{12} + 144 p^{5} T^{13} - 90 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 4 T + 134 T^{2} - 460 T^{3} + 7690 T^{4} - 460 p T^{5} + 134 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 20 T + 192 T^{2} - 28 p T^{3} + 7118 T^{4} - 28 p^{2} T^{5} + 192 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 10 T - 13 T^{2} - 690 T^{3} - 3883 T^{4} - 5440 T^{5} + 46258 T^{6} + 740700 T^{7} + 5790202 T^{8} + 740700 p T^{9} + 46258 p^{2} T^{10} - 5440 p^{3} T^{11} - 3883 p^{4} T^{12} - 690 p^{5} T^{13} - 13 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 - 154 T^{2} + 96 T^{3} + 12457 T^{4} - 9936 T^{5} - 866410 T^{6} + 243120 T^{7} + 52134244 T^{8} + 243120 p T^{9} - 866410 p^{2} T^{10} - 9936 p^{3} T^{11} + 12457 p^{4} T^{12} + 96 p^{5} T^{13} - 154 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 + 6 T - 151 T^{2} - 570 T^{3} + 14677 T^{4} + 25524 T^{5} - 1127266 T^{6} - 657144 T^{7} + 70242250 T^{8} - 657144 p T^{9} - 1127266 p^{2} T^{10} + 25524 p^{3} T^{11} + 14677 p^{4} T^{12} - 570 p^{5} T^{13} - 151 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 12 T^{2} + 768 T^{3} - 2726 T^{4} - 28032 T^{5} + 202320 T^{6} - 539904 T^{7} - 15139581 T^{8} - 539904 p T^{9} + 202320 p^{2} T^{10} - 28032 p^{3} T^{11} - 2726 p^{4} T^{12} + 768 p^{5} T^{13} - 12 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 + 18 T + 155 T^{2} + 1782 T^{3} + 14749 T^{4} + 59688 T^{5} + 736634 T^{6} + 6359652 T^{7} + 26051458 T^{8} + 6359652 p T^{9} + 736634 p^{2} T^{10} + 59688 p^{3} T^{11} + 14749 p^{4} T^{12} + 1782 p^{5} T^{13} + 155 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 42 T + 931 T^{2} - 13278 T^{3} + 132774 T^{4} - 13278 p T^{5} + 931 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 6 T - 201 T^{2} - 1398 T^{3} + 23797 T^{4} + 150156 T^{5} - 1770678 T^{6} - 5160960 T^{7} + 126938826 T^{8} - 5160960 p T^{9} - 1770678 p^{2} T^{10} + 150156 p^{3} T^{11} + 23797 p^{4} T^{12} - 1398 p^{5} T^{13} - 201 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 6 T - 279 T^{2} + 906 T^{3} + 53533 T^{4} - 103164 T^{5} - 6527178 T^{6} + 2673744 T^{7} + 614476002 T^{8} + 2673744 p T^{9} - 6527178 p^{2} T^{10} - 103164 p^{3} T^{11} + 53533 p^{4} T^{12} + 906 p^{5} T^{13} - 279 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 10 T + 241 T^{2} + 2270 T^{3} + 27480 T^{4} + 2270 p T^{5} + 241 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 250 T^{2} + 1008 T^{3} + 34474 T^{4} - 170856 T^{5} - 2791984 T^{6} + 8302896 T^{7} + 205044511 T^{8} + 8302896 p T^{9} - 2791984 p^{2} T^{10} - 170856 p^{3} T^{11} + 34474 p^{4} T^{12} + 1008 p^{5} T^{13} - 250 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 10 T + 363 T^{2} + 2702 T^{3} + 51668 T^{4} + 2702 p T^{5} + 363 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.39619964438396271567063641723, −4.25814966685418883852841273139, −4.03931753715583442161548906214, −3.86145370227337777610812007763, −3.73033545897871746575920757806, −3.72488819047073102709815099706, −3.39354411680838417828043531353, −3.35821958215022254496590548441, −3.35351805398601735874082083054, −2.86527656539661304553450314180, −2.75302580880794212059682203865, −2.65228592326247614451451138429, −2.63681210996530170840361930766, −2.47484280175996843514065575478, −2.08601760294220875165138621527, −2.00812086188858870684426302183, −1.94185073495420425109971288833, −1.48679863622441519156311285902, −1.38338998166535556293017113178, −1.19067730974084589867281068616, −1.04541315029409704312664101336, −0.943199552988592316438381048486, −0.75830558979744123918235720808, −0.55657478244018201094036547041, −0.43884181232487112967781679404,
0.43884181232487112967781679404, 0.55657478244018201094036547041, 0.75830558979744123918235720808, 0.943199552988592316438381048486, 1.04541315029409704312664101336, 1.19067730974084589867281068616, 1.38338998166535556293017113178, 1.48679863622441519156311285902, 1.94185073495420425109971288833, 2.00812086188858870684426302183, 2.08601760294220875165138621527, 2.47484280175996843514065575478, 2.63681210996530170840361930766, 2.65228592326247614451451138429, 2.75302580880794212059682203865, 2.86527656539661304553450314180, 3.35351805398601735874082083054, 3.35821958215022254496590548441, 3.39354411680838417828043531353, 3.72488819047073102709815099706, 3.73033545897871746575920757806, 3.86145370227337777610812007763, 4.03931753715583442161548906214, 4.25814966685418883852841273139, 4.39619964438396271567063641723
Plot not available for L-functions of degree greater than 10.
