| L(s) = 1 | + 9.62·2-s + 8.20·3-s + 60.7·4-s + 25·5-s + 79.0·6-s + 33.4·7-s + 276.·8-s − 175.·9-s + 240.·10-s + 121·11-s + 498.·12-s − 872.·13-s + 322.·14-s + 205.·15-s + 718.·16-s − 1.88e3·17-s − 1.69e3·18-s − 361·19-s + 1.51e3·20-s + 274.·21-s + 1.16e3·22-s − 3.04e3·23-s + 2.26e3·24-s + 625·25-s − 8.40e3·26-s − 3.43e3·27-s + 2.03e3·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.526·3-s + 1.89·4-s + 0.447·5-s + 0.896·6-s + 0.257·7-s + 1.52·8-s − 0.722·9-s + 0.761·10-s + 0.301·11-s + 0.999·12-s − 1.43·13-s + 0.439·14-s + 0.235·15-s + 0.702·16-s − 1.57·17-s − 1.23·18-s − 0.229·19-s + 0.848·20-s + 0.135·21-s + 0.513·22-s − 1.20·23-s + 0.804·24-s + 0.200·25-s − 2.43·26-s − 0.907·27-s + 0.489·28-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 | \( 1 - 9.62T + 32T^{2} \) |
| 3 | \( 1 - 8.20T + 243T^{2} \) |
| 7 | \( 1 - 33.4T + 1.68e4T^{2} \) |
| 13 | \( 1 + 872.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.88e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 10.1T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.34e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.08e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.59e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.23e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.84e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.74e4T + 8.58e9T^{2} \) |
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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
−8.803914358103846802979420562481, −7.68698786233938549359386955935, −6.81849355207895756617529902888, −5.99737460407720268847054385040, −5.20882614960247639533493433801, −4.40048788428186302193215298020, −3.55320926338110112722283103074, −2.33019928721165715923169321170, −2.14829615830897234326097443551, 0,
2.14829615830897234326097443551, 2.33019928721165715923169321170, 3.55320926338110112722283103074, 4.40048788428186302193215298020, 5.20882614960247639533493433801, 5.99737460407720268847054385040, 6.81849355207895756617529902888, 7.68698786233938549359386955935, 8.803914358103846802979420562481
