| L(s) = 1 | + 430·5-s + 1.22e3·7-s + 3.16e3·11-s − 6.11e3·13-s + 1.62e4·17-s − 5.47e3·19-s + 1.57e3·23-s + 1.06e5·25-s + 1.22e5·29-s − 2.51e5·31-s + 5.26e5·35-s + 5.23e4·37-s + 3.19e5·41-s + 7.10e5·43-s + 2.84e5·47-s + 6.74e5·49-s + 2.96e5·53-s + 1.36e6·55-s + 8.97e5·59-s + 8.84e5·61-s − 2.63e6·65-s + 4.65e6·67-s − 2.71e6·71-s − 5.67e6·73-s + 3.87e6·77-s + 5.12e6·79-s + 1.56e6·83-s + ⋯ |
| L(s) = 1 | + 1.53·5-s + 1.34·7-s + 0.716·11-s − 0.772·13-s + 0.803·17-s − 0.183·19-s + 0.0270·23-s + 1.36·25-s + 0.935·29-s − 1.51·31-s + 2.07·35-s + 0.169·37-s + 0.723·41-s + 1.36·43-s + 0.399·47-s + 0.819·49-s + 0.273·53-s + 1.10·55-s + 0.568·59-s + 0.499·61-s − 1.18·65-s + 1.89·67-s − 0.898·71-s − 1.70·73-s + 0.966·77-s + 1.16·79-s + 0.300·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.461871771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.461871771\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 86 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 1224 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3164 T + p^{7} T^{2} \) |
| 13 | \( 1 + 6118 T + p^{7} T^{2} \) |
| 17 | \( 1 - 16270 T + p^{7} T^{2} \) |
| 19 | \( 1 + 5476 T + p^{7} T^{2} \) |
| 23 | \( 1 - 1576 T + p^{7} T^{2} \) |
| 29 | \( 1 - 122838 T + p^{7} T^{2} \) |
| 31 | \( 1 + 251360 T + p^{7} T^{2} \) |
| 37 | \( 1 - 52338 T + p^{7} T^{2} \) |
| 41 | \( 1 - 319398 T + p^{7} T^{2} \) |
| 43 | \( 1 - 710788 T + p^{7} T^{2} \) |
| 47 | \( 1 - 284112 T + p^{7} T^{2} \) |
| 53 | \( 1 - 296062 T + p^{7} T^{2} \) |
| 59 | \( 1 - 897548 T + p^{7} T^{2} \) |
| 61 | \( 1 - 884810 T + p^{7} T^{2} \) |
| 67 | \( 1 - 4659692 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2710792 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5670854 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5124176 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1563556 T + p^{7} T^{2} \) |
| 89 | \( 1 + 11605674 T + p^{7} T^{2} \) |
| 97 | \( 1 - 10931618 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.577746694794394605243501038977, −8.878969116675375605437134511676, −7.83317514767474353323449371536, −6.88938660796647187552506833266, −5.77843077831126401858569610708, −5.20097208480889062503659541077, −4.15290021035455728311869896281, −2.59460824472065701986812311712, −1.77960985790409767400029484248, −0.974238019367193453601253313256,
0.974238019367193453601253313256, 1.77960985790409767400029484248, 2.59460824472065701986812311712, 4.15290021035455728311869896281, 5.20097208480889062503659541077, 5.77843077831126401858569610708, 6.88938660796647187552506833266, 7.83317514767474353323449371536, 8.878969116675375605437134511676, 9.577746694794394605243501038977
