| L(s) = 1 | + 41.4·2-s − 265.·3-s + 1.20e3·4-s − 1.04e3·5-s − 1.10e4·6-s + 2.40e3·7-s + 2.89e4·8-s + 5.07e4·9-s − 4.33e4·10-s + 1.23e4·11-s − 3.20e5·12-s + 2.85e4·13-s + 9.96e4·14-s + 2.77e5·15-s + 5.80e5·16-s − 2.32e5·17-s + 2.10e6·18-s − 4.96e5·19-s − 1.26e6·20-s − 6.37e5·21-s + 5.10e5·22-s − 2.39e6·23-s − 7.67e6·24-s − 8.63e5·25-s + 1.18e6·26-s − 8.25e6·27-s + 2.90e6·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 1.89·3-s + 2.36·4-s − 0.746·5-s − 3.46·6-s + 0.377·7-s + 2.49·8-s + 2.58·9-s − 1.36·10-s + 0.253·11-s − 4.46·12-s + 0.277·13-s + 0.692·14-s + 1.41·15-s + 2.21·16-s − 0.673·17-s + 4.73·18-s − 0.874·19-s − 1.76·20-s − 0.715·21-s + 0.464·22-s − 1.78·23-s − 4.72·24-s − 0.442·25-s + 0.508·26-s − 2.99·27-s + 0.892·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 41.4T + 512T^{2} \) |
| 3 | \( 1 + 265.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.04e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 1.23e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.32e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.96e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.39e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.32e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.00e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.28e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 5.76e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.01e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.29e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.10e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.32e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.97e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.81e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.41e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.96e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.98e8T + 7.60e17T^{2} \) |
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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
−11.67922558080696356098407693078, −11.41634718405986621715321769843, −10.34850923601980423992165931004, −7.62765753736141284061139181617, −6.43764231176036378992862825827, −5.79089548460888998871261100754, −4.53734459853402982257372501592, −4.00571337574243141469448565024, −1.78559481415174726701306132089, 0,
1.78559481415174726701306132089, 4.00571337574243141469448565024, 4.53734459853402982257372501592, 5.79089548460888998871261100754, 6.43764231176036378992862825827, 7.62765753736141284061139181617, 10.34850923601980423992165931004, 11.41634718405986621715321769843, 11.67922558080696356098407693078
