| L(s) = 1 | + 1.11e3·7-s − 1.73e4·13-s + 6.49e4·19-s − 3.14e4·25-s − 4.59e5·31-s − 1.08e6·37-s + 9.30e5·43-s − 7.09e5·49-s − 2.75e5·61-s + 6.28e5·67-s + 5.33e6·73-s − 2.20e6·79-s − 1.93e7·91-s − 5.95e6·97-s + 1.17e7·103-s + 1.78e7·109-s − 1.67e7·121-s + ⋯ |
| L(s) = 1 | + 1.23·7-s − 2.18·13-s + 2.17·19-s − 0.401·25-s − 2.77·31-s − 3.51·37-s + 1.78·43-s − 0.861·49-s − 0.155·61-s + 0.255·67-s + 1.60·73-s − 0.502·79-s − 2.69·91-s − 0.662·97-s + 1.05·103-s + 1.32·109-s − 0.857·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 + 31402 T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 559 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 16713814 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 667 p T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 189231154 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 32461 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18081694 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9599936170 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 229892 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 541177 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 264543016114 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 465112 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 323393423854 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1296296189242 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4360664458486 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 137773 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 314041 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 10295330331310 T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 36569 p T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1101815 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 17317361610166 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 77662517197858 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2979379 T + p^{7} T^{2} )^{2} \) |
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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
−9.552821017205779532158681177861, −9.488184678138110612725020009971, −8.903547739835698161687967322383, −8.407518501929338690248718294940, −7.59437696793553970013952932543, −7.56778069374726642215142674125, −7.23215546116995220514148997904, −6.70002146648205382503499617662, −5.69808272968275851026000524193, −5.39600314721006626806958202866, −4.88914780953385762605793133029, −4.86367293353490437771112098403, −3.66199830440285385174669461066, −3.58153294700245200038100280423, −2.66474831958503940343767385519, −2.10298818601759131331456158027, −1.64680539684650298112342789893, −1.09117204386668636120893799373, 0, 0,
1.09117204386668636120893799373, 1.64680539684650298112342789893, 2.10298818601759131331456158027, 2.66474831958503940343767385519, 3.58153294700245200038100280423, 3.66199830440285385174669461066, 4.86367293353490437771112098403, 4.88914780953385762605793133029, 5.39600314721006626806958202866, 5.69808272968275851026000524193, 6.70002146648205382503499617662, 7.23215546116995220514148997904, 7.56778069374726642215142674125, 7.59437696793553970013952932543, 8.407518501929338690248718294940, 8.903547739835698161687967322383, 9.488184678138110612725020009971, 9.552821017205779532158681177861
