| L(s) = 1 | + 17.8·2-s + 86.1·3-s + 190.·4-s + 250.·5-s + 1.53e3·6-s − 343·7-s + 1.11e3·8-s + 5.23e3·9-s + 4.46e3·10-s − 5.17e3·11-s + 1.64e4·12-s − 2.19e3·13-s − 6.12e3·14-s + 2.15e4·15-s − 4.46e3·16-s − 8.03e3·17-s + 9.35e4·18-s + 1.66e4·19-s + 4.76e4·20-s − 2.95e4·21-s − 9.23e4·22-s − 6.38e4·23-s + 9.61e4·24-s − 1.54e4·25-s − 3.92e4·26-s + 2.63e5·27-s − 6.53e4·28-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.84·3-s + 1.48·4-s + 0.895·5-s + 2.90·6-s − 0.377·7-s + 0.770·8-s + 2.39·9-s + 1.41·10-s − 1.17·11-s + 2.74·12-s − 0.277·13-s − 0.596·14-s + 1.65·15-s − 0.272·16-s − 0.396·17-s + 3.77·18-s + 0.555·19-s + 1.33·20-s − 0.696·21-s − 1.84·22-s − 1.09·23-s + 1.42·24-s − 0.198·25-s − 0.437·26-s + 2.57·27-s − 0.562·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.960239057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.960239057\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 - 17.8T + 128T^{2} \) |
| 3 | \( 1 - 86.1T + 2.18e3T^{2} \) |
| 5 | \( 1 - 250.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 5.17e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 8.03e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.66e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.38e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.07e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.19e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.02e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.48e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.85e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 389.T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.83e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.05e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.48e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.91e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.09e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.22e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.55e7T + 8.07e13T^{2} \) |
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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
−13.14699724262733351583527656969, −12.31239959291191263779397604096, −10.35110975924595521832434036680, −9.411061633847850353648532618822, −8.089227347869507379604507371367, −6.80269363796419591964136439890, −5.36201907344625830135447626130, −3.99659246380238225653981364205, −2.81411005390425253491242693502, −2.13209069277458106650381439216,
2.13209069277458106650381439216, 2.81411005390425253491242693502, 3.99659246380238225653981364205, 5.36201907344625830135447626130, 6.80269363796419591964136439890, 8.089227347869507379604507371367, 9.411061633847850353648532618822, 10.35110975924595521832434036680, 12.31239959291191263779397604096, 13.14699724262733351583527656969
