| L(s) = 1 | − 0.601·2-s − 1.63·4-s + 0.0865·5-s + 1.51·7-s + 2.18·8-s − 0.0520·10-s + 3.05·11-s − 1.94·13-s − 0.913·14-s + 1.96·16-s + 1.67·17-s − 19-s − 0.141·20-s − 1.83·22-s − 3.12·23-s − 4.99·25-s + 1.16·26-s − 2.49·28-s + 6.17·29-s − 6.86·31-s − 5.55·32-s − 1.00·34-s + 0.131·35-s − 2.74·37-s + 0.601·38-s + 0.189·40-s + 0.872·41-s + ⋯ |
| L(s) = 1 | − 0.425·2-s − 0.819·4-s + 0.0387·5-s + 0.574·7-s + 0.773·8-s − 0.0164·10-s + 0.922·11-s − 0.538·13-s − 0.244·14-s + 0.490·16-s + 0.405·17-s − 0.229·19-s − 0.0317·20-s − 0.391·22-s − 0.651·23-s − 0.998·25-s + 0.228·26-s − 0.470·28-s + 1.14·29-s − 1.23·31-s − 0.981·32-s − 0.172·34-s + 0.0222·35-s − 0.450·37-s + 0.0975·38-s + 0.0299·40-s + 0.136·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.601T + 2T^{2} \) |
| 5 | \( 1 - 0.0865T + 5T^{2} \) |
| 7 | \( 1 - 1.51T + 7T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 + 2.74T + 37T^{2} \) |
| 41 | \( 1 - 0.872T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 53 | \( 1 + 7.53T + 53T^{2} \) |
| 59 | \( 1 - 5.19T + 59T^{2} \) |
| 61 | \( 1 + 4.03T + 61T^{2} \) |
| 67 | \( 1 + 3.42T + 67T^{2} \) |
| 71 | \( 1 + 5.05T + 71T^{2} \) |
| 73 | \( 1 + 7.44T + 73T^{2} \) |
| 79 | \( 1 + 9.99T + 79T^{2} \) |
| 83 | \( 1 + 7.74T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 2.11T + 97T^{2} \) |
| 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
−7.64436778917609654970518343566, −6.99594079406305021865875837992, −5.97481703385093640206545182519, −5.44361441460979635229269658208, −4.44018230210522579905532117094, −4.17248405560069180648184324440, −3.16828651450175023236210898375, −1.96032458542871597574102669565, −1.20901462923239291437487342623, 0,
1.20901462923239291437487342623, 1.96032458542871597574102669565, 3.16828651450175023236210898375, 4.17248405560069180648184324440, 4.44018230210522579905532117094, 5.44361441460979635229269658208, 5.97481703385093640206545182519, 6.99594079406305021865875837992, 7.64436778917609654970518343566
