| L(s) = 1 | − 0.836·2-s − 16.2·3-s − 31.3·4-s + 25·5-s + 13.5·6-s − 208.·7-s + 52.9·8-s + 19.6·9-s − 20.9·10-s + 121·11-s + 507.·12-s − 1.12e3·13-s + 174.·14-s − 405.·15-s + 957.·16-s − 340.·17-s − 16.4·18-s − 361·19-s − 782.·20-s + 3.37e3·21-s − 101.·22-s − 1.49e3·23-s − 858.·24-s + 625·25-s + 941.·26-s + 3.61e3·27-s + 6.51e3·28-s + ⋯ |
| L(s) = 1 | − 0.147·2-s − 1.03·3-s − 0.978·4-s + 0.447·5-s + 0.153·6-s − 1.60·7-s + 0.292·8-s + 0.0810·9-s − 0.0661·10-s + 0.301·11-s + 1.01·12-s − 1.84·13-s + 0.237·14-s − 0.464·15-s + 0.934·16-s − 0.285·17-s − 0.0119·18-s − 0.229·19-s − 0.437·20-s + 1.66·21-s − 0.0445·22-s − 0.591·23-s − 0.304·24-s + 0.200·25-s + 0.273·26-s + 0.955·27-s + 1.56·28-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 | \( 1 + 0.836T + 32T^{2} \) |
| 3 | \( 1 + 16.2T + 243T^{2} \) |
| 7 | \( 1 + 208.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.12e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 340.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.12e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.18e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.80e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.85e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−9.122054200924930327740100511748, −7.944506697552230434345128672559, −6.83239460108719266296930358268, −6.20162866573546241236771217751, −5.35057904339225417651590188705, −4.62053929590923523268364668754, −3.49338117920873726065775990554, −2.34478727298973114514481942268, −0.64963293999573086298870848834, 0,
0.64963293999573086298870848834, 2.34478727298973114514481942268, 3.49338117920873726065775990554, 4.62053929590923523268364668754, 5.35057904339225417651590188705, 6.20162866573546241236771217751, 6.83239460108719266296930358268, 7.944506697552230434345128672559, 9.122054200924930327740100511748
