| L(s) = 1 | − 9.20·2-s − 16.1·3-s + 52.7·4-s − 25·5-s + 148.·6-s − 66.9·7-s − 191.·8-s + 18.4·9-s + 230.·10-s + 121·11-s − 853.·12-s + 504.·13-s + 616.·14-s + 404.·15-s + 72.7·16-s − 1.42e3·17-s − 169.·18-s + 361·19-s − 1.31e3·20-s + 1.08e3·21-s − 1.11e3·22-s + 1.50e3·23-s + 3.09e3·24-s + 625·25-s − 4.64e3·26-s + 3.63e3·27-s − 3.53e3·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 1.03·3-s + 1.64·4-s − 0.447·5-s + 1.68·6-s − 0.516·7-s − 1.05·8-s + 0.0759·9-s + 0.727·10-s + 0.301·11-s − 1.71·12-s + 0.827·13-s + 0.840·14-s + 0.463·15-s + 0.0710·16-s − 1.19·17-s − 0.123·18-s + 0.229·19-s − 0.737·20-s + 0.535·21-s − 0.490·22-s + 0.593·23-s + 1.09·24-s + 0.200·25-s − 1.34·26-s + 0.958·27-s − 0.851·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9.20T + 32T^{2} \) |
| 3 | \( 1 + 16.1T + 243T^{2} \) |
| 7 | \( 1 + 66.9T + 1.68e4T^{2} \) |
| 13 | \( 1 - 504.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.42e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.50e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 62.2T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.37e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.53e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.68e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.80e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.33e5T + 8.58e9T^{2} \) |
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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
−8.927878905718990587375506580291, −8.106489478861840374174115803744, −7.08459199526177964586877123205, −6.55312785296354855423392287706, −5.68309770834173577800311178523, −4.45115664353372300387682144945, −3.18955013662177851812605147419, −1.81730299075013830258390759461, −0.73083061147956870457655406326, 0,
0.73083061147956870457655406326, 1.81730299075013830258390759461, 3.18955013662177851812605147419, 4.45115664353372300387682144945, 5.68309770834173577800311178523, 6.55312785296354855423392287706, 7.08459199526177964586877123205, 8.106489478861840374174115803744, 8.927878905718990587375506580291
