| L(s) = 1 | − 3.67·2-s + 5.47·4-s − 0.939·5-s − 4.16·7-s + 9.26·8-s + 3.44·10-s + 62.6·11-s + 67.9·13-s + 15.2·14-s − 77.8·16-s + 84.8·17-s + 15.3·19-s − 5.14·20-s − 229.·22-s − 97.8·23-s − 124.·25-s − 249.·26-s − 22.8·28-s + 57.2·29-s − 125.·31-s + 211.·32-s − 311.·34-s + 3.91·35-s − 210.·37-s − 56.4·38-s − 8.70·40-s − 424.·41-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.684·4-s − 0.0840·5-s − 0.224·7-s + 0.409·8-s + 0.109·10-s + 1.71·11-s + 1.44·13-s + 0.291·14-s − 1.21·16-s + 1.21·17-s + 0.185·19-s − 0.0575·20-s − 2.22·22-s − 0.887·23-s − 0.992·25-s − 1.88·26-s − 0.153·28-s + 0.366·29-s − 0.729·31-s + 1.16·32-s − 1.57·34-s + 0.0188·35-s − 0.933·37-s − 0.240·38-s − 0.0343·40-s − 1.61·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 3.67T + 8T^{2} \) |
| 5 | \( 1 + 0.939T + 125T^{2} \) |
| 7 | \( 1 + 4.16T + 343T^{2} \) |
| 11 | \( 1 - 62.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 15.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 57.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 210.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 424.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 414.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 416.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 221.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 447.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 641.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 204.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 506.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 912.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 579.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 642.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.56e3T + 9.12e5T^{2} \) |
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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
−8.470541477021928241015923402375, −7.85976306244576473478852663287, −6.83539969748207556699501720572, −6.33437003067029185986627123660, −5.28067587183442230616069065456, −3.93682350421298170497089145572, −3.49025055968428238723990095893, −1.70699790696704666120256274872, −1.24873589295000389238177217839, 0,
1.24873589295000389238177217839, 1.70699790696704666120256274872, 3.49025055968428238723990095893, 3.93682350421298170497089145572, 5.28067587183442230616069065456, 6.33437003067029185986627123660, 6.83539969748207556699501720572, 7.85976306244576473478852663287, 8.470541477021928241015923402375
