| L(s) = 1 | + 14.6·2-s − 208.·3-s − 296.·4-s − 1.47e3·5-s − 3.06e3·6-s − 2.18e3·7-s − 1.18e4·8-s + 2.38e4·9-s − 2.17e4·10-s − 4.64e3·11-s + 6.19e4·12-s − 4.32e4·13-s − 3.20e4·14-s + 3.08e5·15-s − 2.22e4·16-s + 1.21e5·17-s + 3.50e5·18-s + 4.79e5·19-s + 4.38e5·20-s + 4.55e5·21-s − 6.81e4·22-s − 1.00e6·23-s + 2.47e6·24-s + 2.34e5·25-s − 6.34e5·26-s − 8.80e5·27-s + 6.46e5·28-s + ⋯ |
| L(s) = 1 | + 0.648·2-s − 1.48·3-s − 0.579·4-s − 1.05·5-s − 0.965·6-s − 0.343·7-s − 1.02·8-s + 1.21·9-s − 0.686·10-s − 0.0956·11-s + 0.862·12-s − 0.419·13-s − 0.222·14-s + 1.57·15-s − 0.0849·16-s + 0.352·17-s + 0.787·18-s + 0.843·19-s + 0.613·20-s + 0.510·21-s − 0.0620·22-s − 0.751·23-s + 1.52·24-s + 0.119·25-s − 0.272·26-s − 0.318·27-s + 0.198·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.4662724714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4662724714\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - 1.87e6T \) |
| good | 2 | \( 1 - 14.6T + 512T^{2} \) |
| 3 | \( 1 + 208.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.47e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.18e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.64e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.32e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.21e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.79e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.00e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.54e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.22e6T + 2.64e13T^{2} \) |
| 41 | \( 1 - 2.45e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.61e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.10e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.60e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.99e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.09e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 5.67e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.34e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.32e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.30e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.74e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.33e9T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.34479486041588526144344976306, −12.77959969606977312848690612473, −12.07860901875758779371841034015, −11.10847899698059110414849112379, −9.554236004561261027032089556184, −7.63625262871896715159551917026, −6.01310855411869053009220355889, −4.92044529743590340231203974527, −3.66032339387979795410456887224, −0.45171193888090354981969275038,
0.45171193888090354981969275038, 3.66032339387979795410456887224, 4.92044529743590340231203974527, 6.01310855411869053009220355889, 7.63625262871896715159551917026, 9.554236004561261027032089556184, 11.10847899698059110414849112379, 12.07860901875758779371841034015, 12.77959969606977312848690612473, 14.34479486041588526144344976306
