| L(s) = 1 | − 7.16·2-s + 27.0·3-s + 19.2·4-s + 25·5-s − 193.·6-s − 115.·7-s + 91.1·8-s + 488.·9-s − 179.·10-s + 121·11-s + 521.·12-s + 69.9·13-s + 826.·14-s + 676.·15-s − 1.26e3·16-s − 1.47e3·17-s − 3.49e3·18-s + 361·19-s + 481.·20-s − 3.12e3·21-s − 866.·22-s − 1.41e3·23-s + 2.46e3·24-s + 625·25-s − 501.·26-s + 6.63e3·27-s − 2.22e3·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 1.73·3-s + 0.602·4-s + 0.447·5-s − 2.19·6-s − 0.890·7-s + 0.503·8-s + 2.01·9-s − 0.566·10-s + 0.301·11-s + 1.04·12-s + 0.114·13-s + 1.12·14-s + 0.775·15-s − 1.23·16-s − 1.23·17-s − 2.54·18-s + 0.229·19-s + 0.269·20-s − 1.54·21-s − 0.381·22-s − 0.559·23-s + 0.873·24-s + 0.200·25-s − 0.145·26-s + 1.75·27-s − 0.536·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.128478045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.128478045\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 7.16T + 32T^{2} \) |
| 3 | \( 1 - 27.0T + 243T^{2} \) |
| 7 | \( 1 + 115.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 69.9T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.47e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.46e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.74e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.22e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.37e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.78e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.25e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.80e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.08e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.10e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.36e5T + 8.58e9T^{2} \) |
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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
−9.147941815893147320966743921365, −8.663777320439910392857289480518, −7.79136803909962167963356468401, −7.09167485419843089141036766210, −6.23144850922111653864003302125, −4.53546370610315101696265436878, −3.63259332129770954025303086608, −2.54005015969193738928779877107, −1.87740837472363264858023420817, −0.69848008647543359279588603143,
0.69848008647543359279588603143, 1.87740837472363264858023420817, 2.54005015969193738928779877107, 3.63259332129770954025303086608, 4.53546370610315101696265436878, 6.23144850922111653864003302125, 7.09167485419843089141036766210, 7.79136803909962167963356468401, 8.663777320439910392857289480518, 9.147941815893147320966743921365
