| L(s) = 1 | − 2.53·2-s − 3.76·3-s − 1.55·4-s + 5·5-s + 9.55·6-s − 9.64·7-s + 24.2·8-s − 12.8·9-s − 12.6·10-s + 11·11-s + 5.86·12-s − 46.9·13-s + 24.4·14-s − 18.8·15-s − 49.1·16-s − 31.6·17-s + 32.5·18-s − 19·19-s − 7.78·20-s + 36.2·21-s − 27.9·22-s + 36.7·23-s − 91.2·24-s + 25·25-s + 119.·26-s + 149.·27-s + 15.0·28-s + ⋯ |
| L(s) = 1 | − 0.897·2-s − 0.724·3-s − 0.194·4-s + 0.447·5-s + 0.649·6-s − 0.520·7-s + 1.07·8-s − 0.475·9-s − 0.401·10-s + 0.301·11-s + 0.140·12-s − 1.00·13-s + 0.467·14-s − 0.323·15-s − 0.767·16-s − 0.451·17-s + 0.426·18-s − 0.229·19-s − 0.0870·20-s + 0.377·21-s − 0.270·22-s + 0.332·23-s − 0.776·24-s + 0.200·25-s + 0.898·26-s + 1.06·27-s + 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3404709530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3404709530\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 2.53T + 8T^{2} \) |
| 3 | \( 1 + 3.76T + 27T^{2} \) |
| 7 | \( 1 + 9.64T + 343T^{2} \) |
| 13 | \( 1 + 46.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.6T + 4.91e3T^{2} \) |
| 23 | \( 1 - 36.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 27.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 85.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 86.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 8.86T + 7.95e4T^{2} \) |
| 47 | \( 1 + 614.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 185.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 347.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 32.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 327.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 567.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 606.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 248.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 669.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 248.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 319.T + 9.12e5T^{2} \) |
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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
−9.558282190252741399722539593669, −8.887815034136643099492595990533, −8.035926260438955081307146329639, −6.99688801570581077511771987064, −6.28575364190438400223728395197, −5.23187704092700969824977428888, −4.51756148647188021964822548274, −3.04832946050041689193194688353, −1.71540412864891009379522443760, −0.36511333633152996622545387513,
0.36511333633152996622545387513, 1.71540412864891009379522443760, 3.04832946050041689193194688353, 4.51756148647188021964822548274, 5.23187704092700969824977428888, 6.28575364190438400223728395197, 6.99688801570581077511771987064, 8.035926260438955081307146329639, 8.887815034136643099492595990533, 9.558282190252741399722539593669
