| L(s) = 1 | − 2-s + 3-s + 4-s + 4.22·5-s − 6-s − 7-s − 8-s + 9-s − 4.22·10-s + 4.87·11-s + 12-s + 0.642·13-s + 14-s + 4.22·15-s + 16-s − 4.22·17-s − 18-s − 3.58·19-s + 4.22·20-s − 21-s − 4.87·22-s + 6.30·23-s − 24-s + 12.8·25-s − 0.642·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.89·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.33·10-s + 1.46·11-s + 0.288·12-s + 0.178·13-s + 0.267·14-s + 1.09·15-s + 0.250·16-s − 1.02·17-s − 0.235·18-s − 0.823·19-s + 0.945·20-s − 0.218·21-s − 1.03·22-s + 1.31·23-s − 0.204·24-s + 2.57·25-s − 0.125·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.129231490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.129231490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 4.22T + 5T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 - 0.642T + 13T^{2} \) |
| 17 | \( 1 + 4.22T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 + 8.76T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 7.24T + 47T^{2} \) |
| 53 | \( 1 - 9.58T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 7.74T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 + 0.789T + 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 9.51T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−9.412411653948227458990452561346, −9.092275201936308125980761881909, −8.580891721443566983897182138654, −7.00641812006070369278443026186, −6.59459694265716908558472022709, −5.83894099567718637845665461046, −4.58572642816542057374062833889, −3.21270725998995033465900631298, −2.16084041468963136460077932284, −1.36407877951953651736421705957,
1.36407877951953651736421705957, 2.16084041468963136460077932284, 3.21270725998995033465900631298, 4.58572642816542057374062833889, 5.83894099567718637845665461046, 6.59459694265716908558472022709, 7.00641812006070369278443026186, 8.580891721443566983897182138654, 9.092275201936308125980761881909, 9.412411653948227458990452561346
