| L(s) = 1 | + 1.85·3-s + 1.44·5-s + 1.96·7-s + 0.453·9-s − 2.85·11-s − 3.84·13-s + 2.68·15-s + 1.30·17-s − 3.53·19-s + 3.65·21-s − 4.20·23-s − 2.91·25-s − 4.73·27-s − 1.10·29-s + 3.53·31-s − 5.30·33-s + 2.83·35-s − 6.61·37-s − 7.14·39-s + 0.671·41-s + 4.32·43-s + 0.653·45-s + 0.378·47-s − 3.13·49-s + 2.42·51-s − 7.13·53-s − 4.12·55-s + ⋯ |
| L(s) = 1 | + 1.07·3-s + 0.645·5-s + 0.742·7-s + 0.151·9-s − 0.861·11-s − 1.06·13-s + 0.692·15-s + 0.316·17-s − 0.810·19-s + 0.796·21-s − 0.877·23-s − 0.583·25-s − 0.910·27-s − 0.205·29-s + 0.634·31-s − 0.923·33-s + 0.479·35-s − 1.08·37-s − 1.14·39-s + 0.104·41-s + 0.659·43-s + 0.0974·45-s + 0.0552·47-s − 0.448·49-s + 0.339·51-s − 0.980·53-s − 0.555·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 1.85T + 3T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 + 3.84T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 + 3.53T + 19T^{2} \) |
| 23 | \( 1 + 4.20T + 23T^{2} \) |
| 29 | \( 1 + 1.10T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 - 0.671T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 - 0.378T + 47T^{2} \) |
| 53 | \( 1 + 7.13T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 6.51T + 61T^{2} \) |
| 67 | \( 1 - 7.55T + 67T^{2} \) |
| 71 | \( 1 + 0.0744T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 + 8.57T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + 8.01T + 97T^{2} \) |
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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
−7.66488720114680914347011068776, −7.00723828939002764931063882298, −5.96697715185238878348302423427, −5.40818575969720963358227364996, −4.64534635112977684771867286561, −3.85315372719958405065143697022, −2.84943288730317361237745264838, −2.28081791294447833927469708487, −1.68386886340034345648761537884, 0,
1.68386886340034345648761537884, 2.28081791294447833927469708487, 2.84943288730317361237745264838, 3.85315372719958405065143697022, 4.64534635112977684771867286561, 5.40818575969720963358227364996, 5.96697715185238878348302423427, 7.00723828939002764931063882298, 7.66488720114680914347011068776
