L(s) = 1 | − 4.92·2-s + 16.7·3-s − 7.73·4-s + 25·5-s − 82.4·6-s − 110.·7-s + 195.·8-s + 37.4·9-s − 123.·10-s + 121·11-s − 129.·12-s + 784.·13-s + 544.·14-s + 418.·15-s − 716.·16-s + 1.58e3·17-s − 184.·18-s − 361·19-s − 193.·20-s − 1.85e3·21-s − 596.·22-s − 4.11e3·23-s + 3.27e3·24-s + 625·25-s − 3.86e3·26-s − 3.44e3·27-s + 854.·28-s + ⋯ |
L(s) = 1 | − 0.870·2-s + 1.07·3-s − 0.241·4-s + 0.447·5-s − 0.935·6-s − 0.852·7-s + 1.08·8-s + 0.154·9-s − 0.389·10-s + 0.301·11-s − 0.259·12-s + 1.28·13-s + 0.742·14-s + 0.480·15-s − 0.699·16-s + 1.33·17-s − 0.134·18-s − 0.229·19-s − 0.108·20-s − 0.915·21-s − 0.262·22-s − 1.62·23-s + 1.16·24-s + 0.200·25-s − 1.12·26-s − 0.908·27-s + 0.206·28-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 | \( 1 + 4.92T + 32T^{2} \) |
| 3 | \( 1 - 16.7T + 243T^{2} \) |
| 7 | \( 1 + 110.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 784.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.58e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.76e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 614.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.85e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.17e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.74e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.07e4T + 8.58e9T^{2} \) |
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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
−8.855150788220278263077827003231, −8.152123668084235649355145360559, −7.53010721004461277722837623486, −6.30128323940648183656184932886, −5.53176446875151997183145982478, −3.93665630512394474280447499874, −3.45953106900920984832764254509, −2.15618227112082296730864731558, −1.22764662144045468697324101204, 0,
1.22764662144045468697324101204, 2.15618227112082296730864731558, 3.45953106900920984832764254509, 3.93665630512394474280447499874, 5.53176446875151997183145982478, 6.30128323940648183656184932886, 7.53010721004461277722837623486, 8.152123668084235649355145360559, 8.855150788220278263077827003231