| L(s) = 1 | − 9.98·3-s − 20.3·5-s + 7·7-s + 72.7·9-s + 11·11-s + 32.5·13-s + 203.·15-s − 125.·17-s + 59.5·19-s − 69.8·21-s + 71.7·23-s + 290.·25-s − 456.·27-s − 299.·29-s − 59.6·31-s − 109.·33-s − 142.·35-s − 130.·37-s − 325.·39-s − 190.·41-s − 164.·43-s − 1.48e3·45-s + 470.·47-s + 49·49-s + 1.25e3·51-s − 290.·53-s − 224.·55-s + ⋯ |
| L(s) = 1 | − 1.92·3-s − 1.82·5-s + 0.377·7-s + 2.69·9-s + 0.301·11-s + 0.694·13-s + 3.50·15-s − 1.78·17-s + 0.718·19-s − 0.726·21-s + 0.650·23-s + 2.32·25-s − 3.25·27-s − 1.91·29-s − 0.345·31-s − 0.579·33-s − 0.689·35-s − 0.580·37-s − 1.33·39-s − 0.724·41-s − 0.583·43-s − 4.91·45-s + 1.45·47-s + 0.142·49-s + 3.43·51-s − 0.752·53-s − 0.549·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3142710712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3142710712\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 9.98T + 27T^{2} \) |
| 5 | \( 1 + 20.3T + 125T^{2} \) |
| 13 | \( 1 - 32.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 125.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 71.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 299.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 59.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 190.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 470.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 373.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 396.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 443.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 563.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 164.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 71.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + 168.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−9.361929886234305407762310419601, −8.438131565182807401049833092607, −7.28221435689571692460458158683, −7.03243179341430423743429687228, −5.97225801145026245009589496686, −4.99787133475834711421373790559, −4.33176734569910017364907589041, −3.61394095699720668112585855971, −1.48758262096974935872103910951, −0.33580283420404299219195422441,
0.33580283420404299219195422441, 1.48758262096974935872103910951, 3.61394095699720668112585855971, 4.33176734569910017364907589041, 4.99787133475834711421373790559, 5.97225801145026245009589496686, 7.03243179341430423743429687228, 7.28221435689571692460458158683, 8.438131565182807401049833092607, 9.361929886234305407762310419601
