| L(s) = 1 | + 17.8·2-s + 189.·4-s + 31.0·5-s − 1.24e3·7-s + 1.10e3·8-s + 553.·10-s + 939.·11-s + 1.01e4·13-s − 2.22e4·14-s − 4.66e3·16-s − 3.39e4·17-s + 1.87e4·19-s + 5.89e3·20-s + 1.67e4·22-s − 1.21e4·23-s − 7.71e4·25-s + 1.80e5·26-s − 2.36e5·28-s − 9.17e4·29-s − 1.87e5·31-s − 2.24e5·32-s − 6.05e5·34-s − 3.87e4·35-s − 1.12e5·37-s + 3.33e5·38-s + 3.41e4·40-s − 2.18e5·41-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.48·4-s + 0.111·5-s − 1.37·7-s + 0.760·8-s + 0.174·10-s + 0.212·11-s + 1.28·13-s − 2.16·14-s − 0.284·16-s − 1.67·17-s + 0.626·19-s + 0.164·20-s + 0.335·22-s − 0.208·23-s − 0.987·25-s + 2.01·26-s − 2.03·28-s − 0.698·29-s − 1.12·31-s − 1.20·32-s − 2.64·34-s − 0.152·35-s − 0.364·37-s + 0.986·38-s + 0.0844·40-s − 0.495·41-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 | \( 1 - 17.8T + 128T^{2} \) |
| 5 | \( 1 - 31.0T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.24e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 939.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.01e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.39e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.87e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 9.17e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.87e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.12e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.18e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.10e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.40e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.35e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.26e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.24e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.30e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 7.40e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.21e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.14e7T + 4.42e13T^{2} \) |
| 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
−11.04934545847610131342396837010, −9.723047774747086958302324534739, −8.714991068199041651226320537540, −6.93887458600869271107565468942, −6.30965392366812325065295618293, −5.39375865557537739891844954289, −3.96217389850277021219682826754, −3.37359029810187515472501938684, −2.01519326045091043598140080430, 0,
2.01519326045091043598140080430, 3.37359029810187515472501938684, 3.96217389850277021219682826754, 5.39375865557537739891844954289, 6.30965392366812325065295618293, 6.93887458600869271107565468942, 8.714991068199041651226320537540, 9.723047774747086958302324534739, 11.04934545847610131342396837010
