| L(s) = 1 | + 4·2-s − 6·3-s + 12·4-s − 24·6-s + 14·7-s + 32·8-s + 27·9-s + 12·11-s − 72·12-s − 100·13-s + 56·14-s + 80·16-s − 80·17-s + 108·18-s − 4·19-s − 84·21-s + 48·22-s − 64·23-s − 192·24-s − 400·26-s − 108·27-s + 168·28-s + 20·29-s − 292·31-s + 192·32-s − 72·33-s − 320·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s + 0.328·11-s − 1.73·12-s − 2.13·13-s + 1.06·14-s + 5/4·16-s − 1.14·17-s + 1.41·18-s − 0.0482·19-s − 0.872·21-s + 0.465·22-s − 0.580·23-s − 1.63·24-s − 3.01·26-s − 0.769·27-s + 1.13·28-s + 0.128·29-s − 1.69·31-s + 1.06·32-s − 0.379·33-s − 1.61·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 12 T + 2314 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 100 T + 5358 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 80 T + 7970 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 13626 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 64 T + 9134 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 20 T + 48782 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 292 T + 80514 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 272 T + 98202 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 132 T + 136054 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 32 T + 41670 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 568 T + 260558 T^{2} + 568 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 348 T + 115966 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 224 T + 30086 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 292622 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 920 T + 667110 T^{2} + 920 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 156 T + 491410 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 332 T + 786774 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 376 T + 1015278 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2016 T + 2094742 T^{2} + 2016 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 548 T + 237398 T^{2} - 548 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1620 T + 2327846 T^{2} + 1620 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.374623529641642991975466465948, −9.137738356446662542975756360178, −8.192058645425010135518549059152, −7.984973711479833271611887670303, −7.34564823190043492990209516901, −7.14547128569767409134490220674, −6.56969653522412311690047296244, −6.42213701692335039407787705438, −5.60332725589983283281187328422, −5.46341527715460912902152918745, −4.87176734727964327441747019391, −4.72481799281693681059016419800, −4.04476067872139953045095260328, −3.95912723624811693238811773981, −2.70796988056929626060748596031, −2.67530130297840793498387480531, −1.59048272111640583740777923326, −1.57719500963848489410822279068, 0, 0,
1.57719500963848489410822279068, 1.59048272111640583740777923326, 2.67530130297840793498387480531, 2.70796988056929626060748596031, 3.95912723624811693238811773981, 4.04476067872139953045095260328, 4.72481799281693681059016419800, 4.87176734727964327441747019391, 5.46341527715460912902152918745, 5.60332725589983283281187328422, 6.42213701692335039407787705438, 6.56969653522412311690047296244, 7.14547128569767409134490220674, 7.34564823190043492990209516901, 7.984973711479833271611887670303, 8.192058645425010135518549059152, 9.137738356446662542975756360178, 9.374623529641642991975466465948
