| L(s) = 1 | + 0.402·2-s − 127.·4-s + 384.·5-s − 88.4·7-s − 102.·8-s + 154.·10-s + 1.54e3·11-s − 1.07e4·13-s − 35.5·14-s + 1.63e4·16-s − 1.33e3·17-s + 2.04e3·19-s − 4.91e4·20-s + 620.·22-s − 1.21e4·23-s + 6.98e4·25-s − 4.30e3·26-s + 1.13e4·28-s − 8.36e4·29-s + 6.48e4·31-s + 1.97e4·32-s − 536.·34-s − 3.40e4·35-s + 4.53e5·37-s + 823.·38-s − 3.95e4·40-s + 3.45e5·41-s + ⋯ |
| L(s) = 1 | + 0.0355·2-s − 0.998·4-s + 1.37·5-s − 0.0974·7-s − 0.0710·8-s + 0.0489·10-s + 0.349·11-s − 1.35·13-s − 0.00346·14-s + 0.996·16-s − 0.0658·17-s + 0.0684·19-s − 1.37·20-s + 0.0124·22-s − 0.208·23-s + 0.894·25-s − 0.0480·26-s + 0.0973·28-s − 0.637·29-s + 0.391·31-s + 0.106·32-s − 0.00234·34-s − 0.134·35-s + 1.47·37-s + 0.00243·38-s − 0.0978·40-s + 0.781·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 0.402T + 128T^{2} \) |
| 5 | \( 1 - 384.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 88.4T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.07e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.33e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.04e3T + 8.93e8T^{2} \) |
| 29 | \( 1 + 8.36e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.48e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.69e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.43e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.85e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.97e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.35e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.47e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.41e7T + 8.07e13T^{2} \) |
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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
−10.18767921902023106655242447491, −9.692144623295186201717763532686, −8.941421234944810118472227220827, −7.65266148418645333802649846097, −6.28557884540861915484665900118, −5.34707409617583810547885698251, −4.38229929648480904240231333762, −2.79986872184595147612812200750, −1.48904159745125467206781965573, 0,
1.48904159745125467206781965573, 2.79986872184595147612812200750, 4.38229929648480904240231333762, 5.34707409617583810547885698251, 6.28557884540861915484665900118, 7.65266148418645333802649846097, 8.941421234944810118472227220827, 9.692144623295186201717763532686, 10.18767921902023106655242447491
