| L(s) = 1 | − 4·2-s + 2·3-s + 12·4-s − 12·5-s − 8·6-s + 18·7-s − 32·8-s − 24·9-s + 48·10-s − 66·11-s + 24·12-s − 10·13-s − 72·14-s − 24·15-s + 80·16-s + 168·17-s + 96·18-s − 42·19-s − 144·20-s + 36·21-s + 264·22-s − 64·24-s − 139·25-s + 40·26-s − 50·27-s + 216·28-s + 210·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.384·3-s + 3/2·4-s − 1.07·5-s − 0.544·6-s + 0.971·7-s − 1.41·8-s − 8/9·9-s + 1.51·10-s − 1.80·11-s + 0.577·12-s − 0.213·13-s − 1.37·14-s − 0.413·15-s + 5/4·16-s + 2.39·17-s + 1.25·18-s − 0.507·19-s − 1.60·20-s + 0.374·21-s + 2.55·22-s − 0.544·24-s − 1.11·25-s + 0.301·26-s − 0.356·27-s + 1.45·28-s + 1.34·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 23 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 28 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 12 T + 283 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 18 T + 620 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 p T + 3508 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 3447 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 168 T + 15910 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 42 T + 14156 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 210 T + 38635 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 32 T + 54546 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 396 T + 78302 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 30 T + 47239 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 696 T + 279818 T^{2} + 696 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 78 T + 176092 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 720 T + 371047 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 774 T + 552724 T^{2} + 774 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 528 T + 303335 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 924 T + 778670 T^{2} + 924 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 786 T + 318148 T^{2} + 786 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 422 T + 393903 T^{2} - 422 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 204 T + 764630 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 888 T + 338458 T^{2} - 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 132 T + 1153219 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 192 T + 1057319 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.177487142290480101550404826457, −8.790878728206466615151164116789, −8.173124090027693510923542171008, −8.128165082102144138743684429652, −7.83298204878190337347516688864, −7.68962543333360823419475969363, −7.17366097791324349829124986384, −6.43485702795086007871888068146, −5.92615627672172256116810405797, −5.57222811855658568766970433300, −4.91428039450126655590437685909, −4.69929998429766719824763150371, −3.65801547522359744739334120899, −3.38869164571305376148983414939, −2.61106097255073073472935746579, −2.52956640085774303411463056388, −1.54770267638401187028531658941, −1.02472591661599781218736351364, 0, 0,
1.02472591661599781218736351364, 1.54770267638401187028531658941, 2.52956640085774303411463056388, 2.61106097255073073472935746579, 3.38869164571305376148983414939, 3.65801547522359744739334120899, 4.69929998429766719824763150371, 4.91428039450126655590437685909, 5.57222811855658568766970433300, 5.92615627672172256116810405797, 6.43485702795086007871888068146, 7.17366097791324349829124986384, 7.68962543333360823419475969363, 7.83298204878190337347516688864, 8.128165082102144138743684429652, 8.173124090027693510923542171008, 8.790878728206466615151164116789, 9.177487142290480101550404826457
