| L(s) = 1 | + 9.38·2-s − 29.8·3-s + 56.0·4-s − 103.·5-s − 279.·6-s − 23.3·7-s + 225.·8-s + 645.·9-s − 971.·10-s − 143.·11-s − 1.66e3·12-s + 18.7·13-s − 219.·14-s + 3.08e3·15-s + 321.·16-s − 680.·17-s + 6.05e3·18-s − 217.·19-s − 5.79e3·20-s + 695.·21-s − 1.34e3·22-s − 2.51e3·23-s − 6.71e3·24-s + 7.58e3·25-s + 176.·26-s − 1.19e4·27-s − 1.30e3·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s − 1.91·3-s + 1.75·4-s − 1.85·5-s − 3.17·6-s − 0.180·7-s + 1.24·8-s + 2.65·9-s − 3.07·10-s − 0.358·11-s − 3.34·12-s + 0.0308·13-s − 0.298·14-s + 3.54·15-s + 0.314·16-s − 0.571·17-s + 4.40·18-s − 0.138·19-s − 3.24·20-s + 0.344·21-s − 0.594·22-s − 0.990·23-s − 2.38·24-s + 2.42·25-s + 0.0511·26-s − 3.16·27-s − 0.315·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 1.36e3T \) |
| good | 2 | \( 1 - 9.38T + 32T^{2} \) |
| 3 | \( 1 + 29.8T + 243T^{2} \) |
| 5 | \( 1 + 103.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 23.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 143.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 18.7T + 3.71e5T^{2} \) |
| 17 | \( 1 + 680.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 217.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.02e3T + 2.86e7T^{2} \) |
| 41 | \( 1 - 8.65e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.80e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.90e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.99e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.01e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.82e4T + 8.58e9T^{2} \) |
| 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
−15.07299839363133557425827378080, −13.09618252254723797785258552693, −12.18910683398517712199783162997, −11.58677294940151259644915656642, −10.74668184973934107972151096333, −7.43266112381139355472735746667, −6.25759601606475927393856056544, −4.85913510688071767076347182400, −3.93481134807616862868337659004, 0,
3.93481134807616862868337659004, 4.85913510688071767076347182400, 6.25759601606475927393856056544, 7.43266112381139355472735746667, 10.74668184973934107972151096333, 11.58677294940151259644915656642, 12.18910683398517712199783162997, 13.09618252254723797785258552693, 15.07299839363133557425827378080
