| L(s) = 1 | − 8·3-s − 10·5-s + 36·7-s + 18·9-s − 22·11-s − 24·13-s + 80·15-s − 8·17-s + 16·19-s − 288·21-s − 312·23-s + 75·25-s + 8·27-s − 284·29-s − 432·31-s + 176·33-s − 360·35-s + 12·37-s + 192·39-s − 164·41-s + 44·43-s − 180·45-s + 152·47-s + 382·49-s + 64·51-s + 124·53-s + 220·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 0.894·5-s + 1.94·7-s + 2/3·9-s − 0.603·11-s − 0.512·13-s + 1.37·15-s − 0.114·17-s + 0.193·19-s − 2.99·21-s − 2.82·23-s + 3/5·25-s + 0.0570·27-s − 1.81·29-s − 2.50·31-s + 0.928·33-s − 1.73·35-s + 0.0533·37-s + 0.788·39-s − 0.624·41-s + 0.156·43-s − 0.596·45-s + 0.471·47-s + 1.11·49-s + 0.175·51-s + 0.321·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 36 T + 914 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 24 T + 3938 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T - 7654 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 16 T + 10326 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 312 T + 48646 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 284 T + 52718 T^{2} + 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 432 T + 104702 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 73598 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 p T + 130742 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 44 T + 158634 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 152 T + 160406 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 124 T + 107198 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1256 T + 745142 T^{2} + 1256 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 788 T + 595374 T^{2} - 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 752 T + 730206 T^{2} + 752 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1312 T + 1035182 T^{2} + 1312 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1480 T + 1322730 T^{2} - 1480 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 40 T + 488814 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1068 T + 1428634 T^{2} - 1068 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 132 T - 745706 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 444 T + 1874534 T^{2} - 444 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.38012735626426728848692193584, −11.36056782078990493670304057768, −10.73016530982524760409826029115, −10.59766310614667589487666983898, −9.734786902558960898690034846266, −9.159491596301138480295185244584, −8.415397637211468723816093433827, −7.961547370348620979134389915276, −7.44030007350764113617321845513, −7.38133655238754834717838909720, −6.14756565494518571034602979677, −5.82203603717048247648731290303, −5.18117432542766668998924888673, −4.93308163648239981291380068464, −4.12326749604103369618165547771, −3.63392329910789445540098333029, −2.15861427808237159644443259167, −1.61080146313856315454010815355, 0, 0,
1.61080146313856315454010815355, 2.15861427808237159644443259167, 3.63392329910789445540098333029, 4.12326749604103369618165547771, 4.93308163648239981291380068464, 5.18117432542766668998924888673, 5.82203603717048247648731290303, 6.14756565494518571034602979677, 7.38133655238754834717838909720, 7.44030007350764113617321845513, 7.961547370348620979134389915276, 8.415397637211468723816093433827, 9.159491596301138480295185244584, 9.734786902558960898690034846266, 10.59766310614667589487666983898, 10.73016530982524760409826029115, 11.36056782078990493670304057768, 11.38012735626426728848692193584
