| L(s) = 1 | − 0.775·2-s − 1.39·4-s − 3.48·5-s + 0.0624·7-s + 2.63·8-s + 2.70·10-s − 3.55·11-s − 0.927·13-s − 0.0484·14-s + 0.756·16-s + 3.32·17-s − 5.02·19-s + 4.87·20-s + 2.75·22-s + 23-s + 7.15·25-s + 0.718·26-s − 0.0873·28-s − 29-s + 4.47·31-s − 5.85·32-s − 2.57·34-s − 0.217·35-s − 1.59·37-s + 3.89·38-s − 9.18·40-s − 1.57·41-s + ⋯ |
| L(s) = 1 | − 0.548·2-s − 0.699·4-s − 1.55·5-s + 0.0236·7-s + 0.931·8-s + 0.854·10-s − 1.07·11-s − 0.257·13-s − 0.0129·14-s + 0.189·16-s + 0.807·17-s − 1.15·19-s + 1.09·20-s + 0.587·22-s + 0.208·23-s + 1.43·25-s + 0.140·26-s − 0.0165·28-s − 0.185·29-s + 0.804·31-s − 1.03·32-s − 0.442·34-s − 0.0368·35-s − 0.261·37-s + 0.632·38-s − 1.45·40-s − 0.245·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.775T + 2T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 7 | \( 1 - 0.0624T + 7T^{2} \) |
| 11 | \( 1 + 3.55T + 11T^{2} \) |
| 13 | \( 1 + 0.927T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 + 5.02T + 19T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 1.59T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 + 2.30T + 43T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 - 9.22T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 7.60T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 8.68T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84381997007728575043073028921, −7.40372229102481646122738507277, −6.50861825276869713506835449677, −5.30886600064037677869810744490, −4.81239563393133065003699119120, −4.01294219185151286696459872678, −3.42129300887846337510019264195, −2.31208011893857025378430736667, −0.858954411071496724526774003616, 0,
0.858954411071496724526774003616, 2.31208011893857025378430736667, 3.42129300887846337510019264195, 4.01294219185151286696459872678, 4.81239563393133065003699119120, 5.30886600064037677869810744490, 6.50861825276869713506835449677, 7.40372229102481646122738507277, 7.84381997007728575043073028921
