| L(s) = 1 | + 4·2-s − 2·3-s + 12·4-s − 8·6-s − 8·7-s + 32·8-s − 24·9-s + 28·11-s − 24·12-s − 2·13-s − 32·14-s + 80·16-s − 152·17-s − 96·18-s − 38·19-s + 16·21-s + 112·22-s − 284·23-s − 64·24-s − 8·26-s + 50·27-s − 96·28-s + 144·29-s + 128·31-s + 192·32-s − 56·33-s − 608·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.384·3-s + 3/2·4-s − 0.544·6-s − 0.431·7-s + 1.41·8-s − 8/9·9-s + 0.767·11-s − 0.577·12-s − 0.0426·13-s − 0.610·14-s + 5/4·16-s − 2.16·17-s − 1.25·18-s − 0.458·19-s + 0.166·21-s + 1.08·22-s − 2.57·23-s − 0.544·24-s − 0.0603·26-s + 0.356·27-s − 0.647·28-s + 0.922·29-s + 0.741·31-s + 1.06·32-s − 0.295·33-s − 3.06·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 28 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8 T + 510 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 28 T + 1658 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 648 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 152 T + 14630 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 284 T + 43910 T^{2} + 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 144 T + 49630 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 128 T + 38286 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 194 T + 104640 T^{2} - 194 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 608 T + 213830 T^{2} - 608 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 288 T + 175862 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 432 T + 188590 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 810 T + 453352 T^{2} + 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 324 T + 333214 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1076 T + 698754 T^{2} + 1076 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 658 T + 661380 T^{2} + 658 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1696 T + 1433198 T^{2} + 1696 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 832 T + 738822 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1428 T + 1222262 T^{2} + 1428 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 508 T + 848942 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 576 T + 448582 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 834 T + 1999160 T^{2} + 834 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.448959636667590691187629706141, −9.087615638466382584048561414803, −8.451647224305470413009014459005, −8.260456193940835627131396145190, −7.59837756726843962762047557598, −7.21978367490010068882005092533, −6.41631033986201676233939676517, −6.28447917937203411615459605031, −6.08687681922474799801208273142, −5.80240638924554050967754806514, −4.67312451042721037812684850559, −4.59794372601258761984655648470, −4.33650191591405446968441595898, −3.65687137884615326326368383470, −2.91833349582925394225778191131, −2.72858653096269150697097643602, −1.91767739377656276507970096672, −1.45978496085265755233525692299, 0, 0,
1.45978496085265755233525692299, 1.91767739377656276507970096672, 2.72858653096269150697097643602, 2.91833349582925394225778191131, 3.65687137884615326326368383470, 4.33650191591405446968441595898, 4.59794372601258761984655648470, 4.67312451042721037812684850559, 5.80240638924554050967754806514, 6.08687681922474799801208273142, 6.28447917937203411615459605031, 6.41631033986201676233939676517, 7.21978367490010068882005092533, 7.59837756726843962762047557598, 8.260456193940835627131396145190, 8.451647224305470413009014459005, 9.087615638466382584048561414803, 9.448959636667590691187629706141
