| L(s) = 1 | + 6·3-s − 14·5-s − 24·7-s + 27·9-s + 22·11-s + 30·13-s − 84·15-s + 106·17-s − 50·19-s − 144·21-s − 134·23-s − 6·25-s + 108·27-s − 198·29-s − 360·31-s + 132·33-s + 336·35-s − 328·37-s + 180·39-s − 782·41-s − 386·43-s − 378·45-s − 266·47-s + 134·49-s + 636·51-s − 522·53-s − 308·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.25·5-s − 1.29·7-s + 9-s + 0.603·11-s + 0.640·13-s − 1.44·15-s + 1.51·17-s − 0.603·19-s − 1.49·21-s − 1.21·23-s − 0.0479·25-s + 0.769·27-s − 1.26·29-s − 2.08·31-s + 0.696·33-s + 1.62·35-s − 1.45·37-s + 0.739·39-s − 2.97·41-s − 1.36·43-s − 1.25·45-s − 0.825·47-s + 0.390·49-s + 1.74·51-s − 1.35·53-s − 0.755·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 14 T + 202 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 24 T + 442 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 30 T + 4522 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 106 T + 7882 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 50 T + 14246 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 134 T + 26398 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 198 T + 57706 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 360 T + 90430 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 328 T + 62630 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 782 T + 285970 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 386 T + 179870 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 266 T + 92542 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 522 T + 295162 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 172 T + 175654 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 778 T + 577250 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 776 T + 528582 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 630 T + 744334 T^{2} + 630 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1296 T + 1178926 T^{2} - 1296 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 652 T + 589506 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 324 T + 579670 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 756 T + 1427110 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 452 T + 982470 T^{2} + 452 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.959088829180883620329534705200, −9.861028590074669395346880996341, −9.321095906434640800250778513516, −8.848569820353239873905411581597, −8.264970101313790304656261171686, −8.204999707146142173908727035681, −7.49956260017542225733054480131, −7.24697802876803776259224523850, −6.55561076875186715878678075720, −6.33732455053718936212720993177, −5.48538108984862071799960021621, −5.03664054129994538883696061922, −3.99360765766063151571840891145, −3.77027883306281974695814571006, −3.31715648920465848974587737425, −3.24809236180737111028778093508, −1.79193001178477313256692615977, −1.66933105027724299302279399416, 0, 0,
1.66933105027724299302279399416, 1.79193001178477313256692615977, 3.24809236180737111028778093508, 3.31715648920465848974587737425, 3.77027883306281974695814571006, 3.99360765766063151571840891145, 5.03664054129994538883696061922, 5.48538108984862071799960021621, 6.33732455053718936212720993177, 6.55561076875186715878678075720, 7.24697802876803776259224523850, 7.49956260017542225733054480131, 8.204999707146142173908727035681, 8.264970101313790304656261171686, 8.848569820353239873905411581597, 9.321095906434640800250778513516, 9.861028590074669395346880996341, 9.959088829180883620329534705200
