| L(s) = 1 | + 4·2-s + 14.7·3-s + 16·4-s − 19.0·5-s + 59.1·6-s + 212.·7-s + 64·8-s − 23.9·9-s − 76.3·10-s − 662.·11-s + 236.·12-s + 1.11e3·13-s + 849.·14-s − 282.·15-s + 256·16-s − 522.·17-s − 95.9·18-s − 361·19-s − 305.·20-s + 3.14e3·21-s − 2.65e3·22-s − 3.21e3·23-s + 947.·24-s − 2.76e3·25-s + 4.44e3·26-s − 3.95e3·27-s + 3.39e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.949·3-s + 0.5·4-s − 0.341·5-s + 0.671·6-s + 1.63·7-s + 0.353·8-s − 0.0987·9-s − 0.241·10-s − 1.65·11-s + 0.474·12-s + 1.82·13-s + 1.15·14-s − 0.324·15-s + 0.250·16-s − 0.438·17-s − 0.0698·18-s − 0.229·19-s − 0.170·20-s + 1.55·21-s − 1.16·22-s − 1.26·23-s + 0.335·24-s − 0.883·25-s + 1.29·26-s − 1.04·27-s + 0.818·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.092212203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.092212203\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 19 | \( 1 + 361T \) |
| good | 3 | \( 1 - 14.7T + 243T^{2} \) |
| 5 | \( 1 + 19.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 212.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 662.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 522.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.21e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.08e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.19e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.51e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.62e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.26e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.56e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.94e4T + 8.58e9T^{2} \) |
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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
−15.17548649197583291338304815093, −13.99032078772933234410068781028, −13.35489327542001597598119056591, −11.61672762616855153689955877291, −10.69122895546949054067932679401, −8.421352315317977076240415353734, −7.87353626410628875190498803639, −5.57548770143407297709892494447, −3.94775800621565179734210641580, −2.15048141749809603558933299848,
2.15048141749809603558933299848, 3.94775800621565179734210641580, 5.57548770143407297709892494447, 7.87353626410628875190498803639, 8.421352315317977076240415353734, 10.69122895546949054067932679401, 11.61672762616855153689955877291, 13.35489327542001597598119056591, 13.99032078772933234410068781028, 15.17548649197583291338304815093
