| L(s) = 1 | − 0.279·3-s − 3.34·5-s − 2.71·7-s − 2.92·9-s + 1.18·11-s + 5.24·13-s + 0.936·15-s + 4.01·17-s + 19-s + 0.758·21-s + 0.661·23-s + 6.21·25-s + 1.65·27-s + 1.52·29-s − 9.62·31-s − 0.332·33-s + 9.08·35-s + 8.93·37-s − 1.46·39-s − 2.27·41-s − 4.20·43-s + 9.78·45-s + 4.37·47-s + 0.366·49-s − 1.12·51-s − 53-s − 3.98·55-s + ⋯ |
| L(s) = 1 | − 0.161·3-s − 1.49·5-s − 1.02·7-s − 0.973·9-s + 0.358·11-s + 1.45·13-s + 0.241·15-s + 0.974·17-s + 0.229·19-s + 0.165·21-s + 0.138·23-s + 1.24·25-s + 0.318·27-s + 0.282·29-s − 1.72·31-s − 0.0578·33-s + 1.53·35-s + 1.46·37-s − 0.234·39-s − 0.355·41-s − 0.640·43-s + 1.45·45-s + 0.638·47-s + 0.0523·49-s − 0.157·51-s − 0.137·53-s − 0.536·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 + 0.279T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 23 | \( 1 - 0.661T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 + 9.62T + 31T^{2} \) |
| 37 | \( 1 - 8.93T + 37T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 + 4.20T + 43T^{2} \) |
| 47 | \( 1 - 4.37T + 47T^{2} \) |
| 59 | \( 1 - 8.44T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 + 4.69T + 71T^{2} \) |
| 73 | \( 1 - 4.92T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 7.19T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 7.42T + 97T^{2} \) |
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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
−8.068535354896494392375084980723, −7.44235736018600046582059570164, −6.55630365440631864621266717778, −5.95975695652349718132376302172, −5.14054379173902984939696830529, −3.85712716558460828761860110297, −3.61664861186100410899529880137, −2.81675583659713293355047188060, −1.09292706349349149468891464573, 0,
1.09292706349349149468891464573, 2.81675583659713293355047188060, 3.61664861186100410899529880137, 3.85712716558460828761860110297, 5.14054379173902984939696830529, 5.95975695652349718132376302172, 6.55630365440631864621266717778, 7.44235736018600046582059570164, 8.068535354896494392375084980723
