| L(s) = 1 | + 1.94·2-s + 1.78·4-s − 0.890·5-s − 3.69·7-s − 0.421·8-s − 1.73·10-s + 4.50·11-s − 2.37·13-s − 7.17·14-s − 4.38·16-s + 8.08·17-s + 3.74·19-s − 1.58·20-s + 8.75·22-s − 23-s − 4.20·25-s − 4.62·26-s − 6.58·28-s + 29-s + 0.830·31-s − 7.68·32-s + 15.7·34-s + 3.28·35-s − 6.72·37-s + 7.28·38-s + 0.375·40-s − 5.54·41-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 0.891·4-s − 0.398·5-s − 1.39·7-s − 0.149·8-s − 0.547·10-s + 1.35·11-s − 0.659·13-s − 1.91·14-s − 1.09·16-s + 1.96·17-s + 0.858·19-s − 0.355·20-s + 1.86·22-s − 0.208·23-s − 0.841·25-s − 0.907·26-s − 1.24·28-s + 0.185·29-s + 0.149·31-s − 1.35·32-s + 2.69·34-s + 0.555·35-s − 1.10·37-s + 1.18·38-s + 0.0594·40-s − 0.866·41-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 | \( 1 - 1.94T + 2T^{2} \) |
| 5 | \( 1 + 0.890T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 - 4.50T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 - 8.08T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 31 | \( 1 - 0.830T + 31T^{2} \) |
| 37 | \( 1 + 6.72T + 37T^{2} \) |
| 41 | \( 1 + 5.54T + 41T^{2} \) |
| 43 | \( 1 + 4.97T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 5.72T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 + 2.66T + 61T^{2} \) |
| 67 | \( 1 + 0.373T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 0.182T + 83T^{2} \) |
| 89 | \( 1 + 0.217T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
| 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
−7.35850076808025687571507900624, −6.86296190653110327476825164685, −6.12898457633779619965159695138, −5.57853440608039834322436315054, −4.80598158457472164409377608577, −3.83752028542601212839229293476, −3.43294643953590124908566988768, −2.91798540377670219941827839406, −1.48696621665579096522315409130, 0,
1.48696621665579096522315409130, 2.91798540377670219941827839406, 3.43294643953590124908566988768, 3.83752028542601212839229293476, 4.80598158457472164409377608577, 5.57853440608039834322436315054, 6.12898457633779619965159695138, 6.86296190653110327476825164685, 7.35850076808025687571507900624
