| L(s) = 1 | + 28.8·2-s − 204.·3-s + 317.·4-s − 258.·5-s − 5.89e3·6-s − 8.86e3·7-s − 5.59e3·8-s + 2.21e4·9-s − 7.45e3·10-s + 3.60e4·11-s − 6.49e4·12-s − 2.85e4·13-s − 2.55e5·14-s + 5.29e4·15-s − 3.23e5·16-s + 3.27e5·17-s + 6.38e5·18-s + 2.65e5·19-s − 8.22e4·20-s + 1.81e6·21-s + 1.03e6·22-s − 2.42e6·23-s + 1.14e6·24-s − 1.88e6·25-s − 8.22e5·26-s − 5.09e5·27-s − 2.81e6·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 1.45·3-s + 0.620·4-s − 0.185·5-s − 1.85·6-s − 1.39·7-s − 0.483·8-s + 1.12·9-s − 0.235·10-s + 0.743·11-s − 0.904·12-s − 0.277·13-s − 1.77·14-s + 0.270·15-s − 1.23·16-s + 0.950·17-s + 1.43·18-s + 0.467·19-s − 0.114·20-s + 2.03·21-s + 0.946·22-s − 1.80·23-s + 0.704·24-s − 0.965·25-s − 0.353·26-s − 0.184·27-s − 0.865·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 28.8T + 512T^{2} \) |
| 3 | \( 1 + 204.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 258.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.86e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.60e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 3.27e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.42e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.99e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.45e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.15e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.20e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.13e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.89e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.28e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.91e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.40e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.19e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.44e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.90e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.65e8T + 7.60e17T^{2} \) |
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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
−16.69703621810807532671241614519, −15.72781718784338550861644124803, −13.95928460430890317870092855744, −12.43626443260267289556638154263, −11.83918470565123820424209471048, −9.892948414715274431333698709278, −6.57639634467340707158950518251, −5.54522832179349602028947311014, −3.73592644965051715793501526344, 0,
3.73592644965051715793501526344, 5.54522832179349602028947311014, 6.57639634467340707158950518251, 9.892948414715274431333698709278, 11.83918470565123820424209471048, 12.43626443260267289556638154263, 13.95928460430890317870092855744, 15.72781718784338550861644124803, 16.69703621810807532671241614519
