| L(s) = 1 | − 2-s + 5·3-s − 5·4-s − 5·6-s − 14·7-s + 3·8-s − 25·9-s − 56·11-s − 25·12-s − 52·13-s + 14·14-s − 29·16-s + 103·17-s + 25·18-s − 57·19-s − 70·21-s + 56·22-s + 31·23-s + 15·24-s + 52·26-s − 240·27-s + 70·28-s − 413·29-s − 162·31-s + 115·32-s − 280·33-s − 103·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.962·3-s − 5/8·4-s − 0.340·6-s − 0.755·7-s + 0.132·8-s − 0.925·9-s − 1.53·11-s − 0.601·12-s − 1.10·13-s + 0.267·14-s − 0.453·16-s + 1.46·17-s + 0.327·18-s − 0.688·19-s − 0.727·21-s + 0.542·22-s + 0.281·23-s + 0.127·24-s + 0.392·26-s − 1.71·27-s + 0.472·28-s − 2.64·29-s − 0.938·31-s + 0.635·32-s − 1.47·33-s − 0.519·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 p T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 5 T + 50 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 56 T + 2421 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 p T + 4414 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 103 T + 12222 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 p T + 14028 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 31 T + 24318 T^{2} - 31 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 413 T + 83948 T^{2} + 413 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 162 T + 21494 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 75 T + 69410 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 505 T + 163458 T^{2} + 505 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 73 T + 99574 T^{2} - 73 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 224 T + 100634 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 262 T + 239106 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 190 T - 32242 T^{2} - 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 990 T + 669098 T^{2} - 990 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 908 T + 525193 T^{2} + 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 127 T - 221188 T^{2} - 127 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 337 T + 64726 T^{2} - 337 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1119 T + 1281888 T^{2} + 1119 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1517 T + 1670096 T^{2} - 1517 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1713 T + 2140568 T^{2} + 1713 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1764 T + 1844934 T^{2} + 1764 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.12051641293368434184237837492, −11.41832470961225980992717063400, −11.01536046124060297222088543429, −10.27225130164580296470587129842, −9.682786504174133975214554776520, −9.676777644196231857586231896516, −8.811989879933023661582142136227, −8.573381728798433632334930724805, −8.030839375156797704516647440047, −7.42019505391068780583462265063, −7.08053852838920900218055868615, −5.99964463771112478201905188201, −5.26334827648799051867220881762, −5.24844023302266390233237627885, −3.91830347509576499915907771365, −3.34707444407198510518869819137, −2.69632348137062077335168125393, −2.05284208090969814397299941670, 0, 0,
2.05284208090969814397299941670, 2.69632348137062077335168125393, 3.34707444407198510518869819137, 3.91830347509576499915907771365, 5.24844023302266390233237627885, 5.26334827648799051867220881762, 5.99964463771112478201905188201, 7.08053852838920900218055868615, 7.42019505391068780583462265063, 8.030839375156797704516647440047, 8.573381728798433632334930724805, 8.811989879933023661582142136227, 9.676777644196231857586231896516, 9.682786504174133975214554776520, 10.27225130164580296470587129842, 11.01536046124060297222088543429, 11.41832470961225980992717063400, 12.12051641293368434184237837492
