| L(s) = 1 | + 20.2·2-s + 719.·3-s − 1.63e3·4-s + 1.20e4·5-s + 1.46e4·6-s + 8.77e3·7-s − 7.47e4·8-s + 3.40e5·9-s + 2.44e5·10-s + 4.97e4·11-s − 1.17e6·12-s + 7.43e5·13-s + 1.78e5·14-s + 8.66e6·15-s + 1.83e6·16-s − 1.09e6·17-s + 6.91e6·18-s − 1.83e7·19-s − 1.96e7·20-s + 6.31e6·21-s + 1.00e6·22-s + 1.17e7·23-s − 5.38e7·24-s + 9.60e7·25-s + 1.50e7·26-s + 1.17e8·27-s − 1.43e7·28-s + ⋯ |
| L(s) = 1 | + 0.448·2-s + 1.71·3-s − 0.798·4-s + 1.72·5-s + 0.766·6-s + 0.197·7-s − 0.806·8-s + 1.92·9-s + 0.772·10-s + 0.0931·11-s − 1.36·12-s + 0.555·13-s + 0.0884·14-s + 2.94·15-s + 0.437·16-s − 0.186·17-s + 0.862·18-s − 1.70·19-s − 1.37·20-s + 0.337·21-s + 0.0417·22-s + 0.380·23-s − 1.37·24-s + 1.96·25-s + 0.249·26-s + 1.58·27-s − 0.157·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.728419131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.728419131\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 - 20.2T + 2.04e3T^{2} \) |
| 3 | \( 1 - 719.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 1.20e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 8.77e3T + 1.97e9T^{2} \) |
| 11 | \( 1 - 4.97e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 7.43e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.09e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.83e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.17e7T + 9.52e14T^{2} \) |
| 31 | \( 1 - 1.71e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.53e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 2.71e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.58e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.95e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.92e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 7.96e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 1.26e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.20e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 7.30e7T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.05e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 9.55e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.38e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.67e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.22e11T + 7.15e21T^{2} \) |
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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
−14.32707154986811993797394247908, −13.45471942166520475702882237722, −12.97602509175531034442565422041, −10.19249784504899408051367577255, −9.183750780872166816678299694558, −8.416583061816060311672628305504, −6.24679119100635358615233034861, −4.49930229697622381040243872156, −2.91279082303331839620494981245, −1.65531104080260233065011278421,
1.65531104080260233065011278421, 2.91279082303331839620494981245, 4.49930229697622381040243872156, 6.24679119100635358615233034861, 8.416583061816060311672628305504, 9.183750780872166816678299694558, 10.19249784504899408051367577255, 12.97602509175531034442565422041, 13.45471942166520475702882237722, 14.32707154986811993797394247908
