| L(s) = 1 | + 41.4·2-s − 119.·3-s + 1.20e3·4-s + 1.15e3·5-s − 4.95e3·6-s − 2.40e3·7-s + 2.86e4·8-s − 5.39e3·9-s + 4.78e4·10-s + 9.30e4·11-s − 1.43e5·12-s + 2.85e4·13-s − 9.94e4·14-s − 1.37e5·15-s + 5.70e5·16-s − 4.71e5·17-s − 2.23e5·18-s + 6.78e5·19-s + 1.38e6·20-s + 2.86e5·21-s + 3.85e6·22-s + 8.62e5·23-s − 3.42e6·24-s − 6.20e5·25-s + 1.18e6·26-s + 2.99e6·27-s − 2.89e6·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 0.851·3-s + 2.35·4-s + 0.825·5-s − 1.55·6-s − 0.377·7-s + 2.47·8-s − 0.274·9-s + 1.51·10-s + 1.91·11-s − 2.00·12-s + 0.277·13-s − 0.691·14-s − 0.703·15-s + 2.17·16-s − 1.36·17-s − 0.501·18-s + 1.19·19-s + 1.94·20-s + 0.322·21-s + 3.50·22-s + 0.642·23-s − 2.10·24-s − 0.317·25-s + 0.507·26-s + 1.08·27-s − 0.888·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.177449589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.177449589\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 41.4T + 512T^{2} \) |
| 3 | \( 1 + 119.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.15e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 9.30e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 4.71e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.78e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 8.62e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.95e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.03e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.81e4T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.26e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.76e5T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.78e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.11e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.24e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 9.74e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.73e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.07e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.69e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.82e6T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.40e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.04e8T + 7.60e17T^{2} \) |
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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
−12.13586423387294081219284803907, −11.72053419387122423695445446345, −10.64037800273928695014243146693, −9.121030906619142178393301699793, −6.69961169908448007167166585557, −6.33669999903527392399127007078, −5.28435350250984927688149528723, −4.14659592552071286084817126071, −2.79554341521305103198610467295, −1.26654040562222439126315726224,
1.26654040562222439126315726224, 2.79554341521305103198610467295, 4.14659592552071286084817126071, 5.28435350250984927688149528723, 6.33669999903527392399127007078, 6.69961169908448007167166585557, 9.121030906619142178393301699793, 10.64037800273928695014243146693, 11.72053419387122423695445446345, 12.13586423387294081219284803907
