| L(s) = 1 | + 29.4·2-s − 172.·3-s + 352.·4-s + 865.·5-s − 5.06e3·6-s + 362.·7-s − 4.67e3·8-s + 9.98e3·9-s + 2.54e4·10-s − 2.58e3·11-s − 6.07e4·12-s + 1.51e5·13-s + 1.06e4·14-s − 1.49e5·15-s − 3.18e5·16-s + 2.93e5·18-s − 6.14e5·19-s + 3.05e5·20-s − 6.23e4·21-s − 7.60e4·22-s − 9.25e4·23-s + 8.05e5·24-s − 1.20e6·25-s + 4.46e6·26-s + 1.67e6·27-s + 1.27e5·28-s + 4.01e6·29-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 1.22·3-s + 0.689·4-s + 0.619·5-s − 1.59·6-s + 0.0570·7-s − 0.403·8-s + 0.507·9-s + 0.805·10-s − 0.0532·11-s − 0.846·12-s + 1.47·13-s + 0.0741·14-s − 0.760·15-s − 1.21·16-s + 0.659·18-s − 1.08·19-s + 0.426·20-s − 0.0700·21-s − 0.0691·22-s − 0.0689·23-s + 0.495·24-s − 0.616·25-s + 1.91·26-s + 0.605·27-s + 0.0393·28-s + 1.05·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.670686068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.670686068\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 29.4T + 512T^{2} \) |
| 3 | \( 1 + 172.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 865.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 362.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.58e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.51e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 6.14e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 9.25e4T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.01e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.53e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.19e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.69e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.73e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.76e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.42e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.39e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.98e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.46e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.68e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.27e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.96e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.63e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.04e8T + 7.60e17T^{2} \) |
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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
−10.76866437426702523614850274253, −9.426069570000244243733521169839, −8.290555307782785157306739124917, −6.47032674662220290676430789222, −6.20430161006970191792926547618, −5.32950591753333004291629489556, −4.46159654002851542879459708360, −3.39767400278521836815271611083, −2.00069803329177686252501348911, −0.62690739114490293528887677848,
0.62690739114490293528887677848, 2.00069803329177686252501348911, 3.39767400278521836815271611083, 4.46159654002851542879459708360, 5.32950591753333004291629489556, 6.20430161006970191792926547618, 6.47032674662220290676430789222, 8.290555307782785157306739124917, 9.426069570000244243733521169839, 10.76866437426702523614850274253
