| L(s) = 1 | − 7.96·2-s + 25.0·3-s + 31.3·4-s − 25·5-s − 199.·6-s − 85.0·7-s + 4.94·8-s + 382.·9-s + 199.·10-s + 121·11-s + 784.·12-s − 397.·13-s + 677.·14-s − 625.·15-s − 1.04e3·16-s + 1.04e3·17-s − 3.04e3·18-s − 361·19-s − 784.·20-s − 2.12e3·21-s − 963.·22-s − 1.87e3·23-s + 123.·24-s + 625·25-s + 3.16e3·26-s + 3.48e3·27-s − 2.66e3·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 1.60·3-s + 0.980·4-s − 0.447·5-s − 2.25·6-s − 0.656·7-s + 0.0273·8-s + 1.57·9-s + 0.629·10-s + 0.301·11-s + 1.57·12-s − 0.652·13-s + 0.923·14-s − 0.717·15-s − 1.01·16-s + 0.880·17-s − 2.21·18-s − 0.229·19-s − 0.438·20-s − 1.05·21-s − 0.424·22-s − 0.737·23-s + 0.0438·24-s + 0.200·25-s + 0.918·26-s + 0.918·27-s − 0.643·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.427971539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427971539\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 7.96T + 32T^{2} \) |
| 3 | \( 1 - 25.0T + 243T^{2} \) |
| 7 | \( 1 + 85.0T + 1.68e4T^{2} \) |
| 13 | \( 1 + 397.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.04e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.87e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.00e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.41e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.82e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.81e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.71e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.22e4T + 8.58e9T^{2} \) |
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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
−9.051335671192676865437800324441, −8.541074821479253972216062004755, −7.60886965515050230436331909761, −7.42814470227875501469101773425, −6.20285588228024058309727229193, −4.52404821329418278753318326343, −3.58088270339741500237771209739, −2.67014840434283859346647899384, −1.75359360264433867476706354796, −0.58805885505125209338606539123,
0.58805885505125209338606539123, 1.75359360264433867476706354796, 2.67014840434283859346647899384, 3.58088270339741500237771209739, 4.52404821329418278753318326343, 6.20285588228024058309727229193, 7.42814470227875501469101773425, 7.60886965515050230436331909761, 8.541074821479253972216062004755, 9.051335671192676865437800324441
