| L(s) = 1 | − 9.80·2-s − 29.7·3-s + 64.1·4-s + 25·5-s + 291.·6-s − 68.7·7-s − 315.·8-s + 639.·9-s − 245.·10-s − 393.·11-s − 1.90e3·12-s − 251.·13-s + 674.·14-s − 742.·15-s + 1.04e3·16-s − 1.00e3·17-s − 6.26e3·18-s − 2.74e3·19-s + 1.60e3·20-s + 2.04e3·21-s + 3.86e3·22-s − 529·23-s + 9.38e3·24-s + 625·25-s + 2.47e3·26-s − 1.17e4·27-s − 4.41e3·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 1.90·3-s + 2.00·4-s + 0.447·5-s + 3.30·6-s − 0.530·7-s − 1.74·8-s + 2.63·9-s − 0.775·10-s − 0.981·11-s − 3.82·12-s − 0.413·13-s + 0.920·14-s − 0.852·15-s + 1.01·16-s − 0.839·17-s − 4.56·18-s − 1.74·19-s + 0.897·20-s + 1.01·21-s + 1.70·22-s − 0.208·23-s + 3.32·24-s + 0.200·25-s + 0.716·26-s − 3.10·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1326163429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1326163429\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 9.80T + 32T^{2} \) |
| 3 | \( 1 + 29.7T + 243T^{2} \) |
| 7 | \( 1 + 68.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 393.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 251.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.00e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.74e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 7.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.62e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.68e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.78e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.23e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.78e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.34e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.40e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.76e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.19e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.61e4T + 8.58e9T^{2} \) |
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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
−12.23990374550429577118617599049, −11.10016893886584570245541206259, −10.41773790582281340565919886952, −9.861291601036216858920320449009, −8.401584041663148854892214695044, −6.88791215358297239334292175587, −6.36669350676329769701515461344, −4.92840392119032857193687361694, −1.97695302924953878943791026876, −0.34493055359336298149029354085,
0.34493055359336298149029354085, 1.97695302924953878943791026876, 4.92840392119032857193687361694, 6.36669350676329769701515461344, 6.88791215358297239334292175587, 8.401584041663148854892214695044, 9.861291601036216858920320449009, 10.41773790582281340565919886952, 11.10016893886584570245541206259, 12.23990374550429577118617599049
