| L(s) = 1 | + 4·2-s + 12·4-s + 12·5-s + 16·7-s + 32·8-s + 48·10-s + 36·11-s + 10·13-s + 64·14-s + 80·16-s + 120·17-s − 8·19-s + 144·20-s + 144·22-s + 180·23-s − 115·25-s + 40·26-s + 192·28-s + 324·29-s − 248·31-s + 192·32-s + 480·34-s + 192·35-s − 434·37-s − 32·38-s + 384·40-s + 192·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.07·5-s + 0.863·7-s + 1.41·8-s + 1.51·10-s + 0.986·11-s + 0.213·13-s + 1.22·14-s + 5/4·16-s + 1.71·17-s − 0.0965·19-s + 1.60·20-s + 1.39·22-s + 1.63·23-s − 0.919·25-s + 0.301·26-s + 1.29·28-s + 2.07·29-s − 1.43·31-s + 1.06·32-s + 2.42·34-s + 0.927·35-s − 1.92·37-s − 0.136·38-s + 1.51·40-s + 0.731·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.992250634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.992250634\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 12 T + 259 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 642 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 36 T + 1258 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 T - 873 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 120 T + 10159 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 666 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 180 T + 32002 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 324 T + 73699 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 p T + 39966 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 434 T + 148287 T^{2} + 434 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 192 T + 94786 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 608 T + 250458 T^{2} + 608 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 384 T + 209518 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 408 T + 183418 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1008 T + 637126 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 742 T + 9555 p T^{2} - 742 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 104 T + 90042 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1140 T + 1038994 T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 850 T + 709827 T^{2} - 850 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 440 T + 966978 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 264 T + 1063798 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 768 T + 1343527 T^{2} - 768 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 764 T + 1886598 T^{2} + 764 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.47515402996020006997704812855, −12.42283032257589830424946668410, −11.63403977866921257216211846499, −11.48066686533416006132034602979, −10.69286898278130442449776872695, −10.35093207797022320730613229629, −9.638795059954028756377581521593, −9.277693207317947301551367162184, −8.309729674043154506169870051319, −8.012847443392057408370209280207, −7.08841799332432652433598016869, −6.66083981170901089288911655180, −6.10597376771251412419176777401, −5.39133712422989341172735431005, −5.10071209293664612680295204910, −4.43727610114399171016454761723, −3.43062716479845134539794158742, −3.07815256205359778429489881581, −1.66522117480106884789411934915, −1.49907660947613587642854628404,
1.49907660947613587642854628404, 1.66522117480106884789411934915, 3.07815256205359778429489881581, 3.43062716479845134539794158742, 4.43727610114399171016454761723, 5.10071209293664612680295204910, 5.39133712422989341172735431005, 6.10597376771251412419176777401, 6.66083981170901089288911655180, 7.08841799332432652433598016869, 8.012847443392057408370209280207, 8.309729674043154506169870051319, 9.277693207317947301551367162184, 9.638795059954028756377581521593, 10.35093207797022320730613229629, 10.69286898278130442449776872695, 11.48066686533416006132034602979, 11.63403977866921257216211846499, 12.42283032257589830424946668410, 12.47515402996020006997704812855
