| L(s) = 1 | − 4·2-s + 6·3-s + 12·4-s − 10·5-s − 24·6-s − 8·7-s − 32·8-s + 27·9-s + 40·10-s + 12·11-s + 72·12-s − 36·13-s + 32·14-s − 60·15-s + 80·16-s + 56·17-s − 108·18-s − 38·19-s − 120·20-s − 48·21-s − 48·22-s + 52·23-s − 192·24-s + 75·25-s + 144·26-s + 108·27-s − 96·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 0.431·7-s − 1.41·8-s + 9-s + 1.26·10-s + 0.328·11-s + 1.73·12-s − 0.768·13-s + 0.610·14-s − 1.03·15-s + 5/4·16-s + 0.798·17-s − 1.41·18-s − 0.458·19-s − 1.34·20-s − 0.498·21-s − 0.465·22-s + 0.471·23-s − 1.63·24-s + 3/5·25-s + 1.08·26-s + 0.769·27-s − 0.647·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 8 T + 696 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 12 T + 2548 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 36 T + 4568 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 56 T + 5210 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 52 T - 1126 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 16 T + 1316 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 508 T + 123882 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 68 T + 91368 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 352 T + 138572 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 432 T + 194576 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 204 T + 97066 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 72 T - 164654 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 376 T + 356798 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 112 T + 325674 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 496 T + 662934 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1176 T + 853990 T^{2} - 1176 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 396 T + 291542 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1520 T + 1459134 T^{2} + 1520 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 244 T + 1019834 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1936 T + 2128076 T^{2} + 1936 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1092 T + 1649696 T^{2} + 1092 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.01065440888076264751039042480, −9.535186562975649766768640343717, −9.150881277646781475219816618254, −8.912165767934093657512940114550, −8.191676573764792397515861563107, −8.166103619588675679681008980205, −7.46398185196033245445784444636, −7.30570599021446667270062282152, −6.65823851905920642918055645831, −6.51032508156922450891098531684, −5.29924316580971705839197354430, −5.19951601282060822271943882571, −3.95001326679354123309430185028, −3.84696947372065066529549721253, −2.93320104786111312345537079022, −2.85313727177227370215525213647, −1.58867436097291522064305722183, −1.55822402272918291515946898507, 0, 0,
1.55822402272918291515946898507, 1.58867436097291522064305722183, 2.85313727177227370215525213647, 2.93320104786111312345537079022, 3.84696947372065066529549721253, 3.95001326679354123309430185028, 5.19951601282060822271943882571, 5.29924316580971705839197354430, 6.51032508156922450891098531684, 6.65823851905920642918055645831, 7.30570599021446667270062282152, 7.46398185196033245445784444636, 8.166103619588675679681008980205, 8.191676573764792397515861563107, 8.912165767934093657512940114550, 9.150881277646781475219816618254, 9.535186562975649766768640343717, 10.01065440888076264751039042480
