| L(s) = 1 | − 8·5-s + 16·7-s + 8·11-s − 72·13-s + 36·17-s + 136·19-s − 256·23-s + 6·25-s + 152·29-s − 80·31-s − 128·35-s − 136·37-s + 436·41-s − 712·43-s + 224·47-s − 286·49-s + 344·53-s − 64·55-s + 648·59-s − 648·61-s + 576·65-s + 456·67-s − 2.04e3·71-s + 660·73-s + 128·77-s + 496·79-s − 776·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s + 0.863·7-s + 0.219·11-s − 1.53·13-s + 0.513·17-s + 1.64·19-s − 2.32·23-s + 0.0479·25-s + 0.973·29-s − 0.463·31-s − 0.618·35-s − 0.604·37-s + 1.66·41-s − 2.52·43-s + 0.695·47-s − 0.833·49-s + 0.891·53-s − 0.156·55-s + 1.42·59-s − 1.36·61-s + 1.09·65-s + 0.831·67-s − 3.42·71-s + 1.05·73-s + 0.189·77-s + 0.706·79-s − 1.02·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 542 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 8 T - 650 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 72 T + 4858 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 36 T + 6822 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 136 T + 15014 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 256 T + 39886 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 152 T + 37706 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 80 T + 26030 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 136 T + 102602 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 436 T + 102166 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 712 T + 282422 T^{2} + 712 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 224 T + 179422 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 344 T + 280538 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 324 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 324 T + p^{3} T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 456 T + 174278 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2048 T + 1763566 T^{2} + 2048 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 660 T + 407702 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 496 T + 323534 T^{2} - 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 776 T + 1131046 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 532 T + 1467382 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1220 T + 586694 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.321340917760891867023696257557, −8.112571147768108860484412979614, −7.68192183914758571405713469585, −7.36815772976542513667876137099, −7.07922027094612274008358936879, −6.69471863740239650387205305697, −5.89522538231772232621084453372, −5.75441572591737456190509736066, −5.27604109642396013829914516997, −4.80541742614117770245747189644, −4.45053817630798894964057847723, −4.13813655066905111262332632306, −3.35615889906124741080397609818, −3.31694310577888423923807076601, −2.45135219611688587273639049675, −2.14659462555363910263270581179, −1.42575279509124588216066476077, −1.04050503978164059577705528502, 0, 0,
1.04050503978164059577705528502, 1.42575279509124588216066476077, 2.14659462555363910263270581179, 2.45135219611688587273639049675, 3.31694310577888423923807076601, 3.35615889906124741080397609818, 4.13813655066905111262332632306, 4.45053817630798894964057847723, 4.80541742614117770245747189644, 5.27604109642396013829914516997, 5.75441572591737456190509736066, 5.89522538231772232621084453372, 6.69471863740239650387205305697, 7.07922027094612274008358936879, 7.36815772976542513667876137099, 7.68192183914758571405713469585, 8.112571147768108860484412979614, 8.321340917760891867023696257557
