| L(s) = 1 | − 1.94·2-s − 4.22·4-s − 4.10·5-s + 7·7-s + 23.7·8-s + 7.96·10-s + 11·11-s + 71.7·13-s − 13.5·14-s − 12.3·16-s + 79.3·17-s − 159.·19-s + 17.3·20-s − 21.3·22-s − 137.·23-s − 108.·25-s − 139.·26-s − 29.5·28-s + 72.2·29-s + 235.·31-s − 166.·32-s − 154.·34-s − 28.7·35-s − 236.·37-s + 310.·38-s − 97.4·40-s − 147.·41-s + ⋯ |
| L(s) = 1 | − 0.686·2-s − 0.528·4-s − 0.366·5-s + 0.377·7-s + 1.04·8-s + 0.251·10-s + 0.301·11-s + 1.53·13-s − 0.259·14-s − 0.192·16-s + 1.13·17-s − 1.93·19-s + 0.193·20-s − 0.207·22-s − 1.25·23-s − 0.865·25-s − 1.05·26-s − 0.199·28-s + 0.462·29-s + 1.36·31-s − 0.917·32-s − 0.777·34-s − 0.138·35-s − 1.05·37-s + 1.32·38-s − 0.385·40-s − 0.562·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.102556349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102556349\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 1.94T + 8T^{2} \) |
| 5 | \( 1 + 4.10T + 125T^{2} \) |
| 13 | \( 1 - 71.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 79.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 72.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 235.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 236.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 538.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 571.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 749.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 758.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 40.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 131.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 238.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 588.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 481.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 667.T + 9.12e5T^{2} \) |
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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
−10.24095027104907296538447776596, −8.985807989265438122613708101813, −8.354130723611352440966111069775, −7.888930492273827740534429886831, −6.58909122142811947139107185380, −5.63165356491085443003866395235, −4.31472220197792158485871813866, −3.73029386484504597006288422349, −1.86405085266360553966922247707, −0.70130501464473982058510557113,
0.70130501464473982058510557113, 1.86405085266360553966922247707, 3.73029386484504597006288422349, 4.31472220197792158485871813866, 5.63165356491085443003866395235, 6.58909122142811947139107185380, 7.888930492273827740534429886831, 8.354130723611352440966111069775, 8.985807989265438122613708101813, 10.24095027104907296538447776596
