| L(s) = 1 | − 1.33·3-s − 1.76·5-s − 1.22·9-s − 2.48·11-s + 2.30·13-s + 2.35·15-s + 1.40·17-s − 3.95·19-s − 0.0405·23-s − 1.87·25-s + 5.62·27-s + 6.20·29-s + 2.98·31-s + 3.31·33-s + 0.195·37-s − 3.06·39-s − 41-s − 1.19·43-s + 2.15·45-s + 7.43·47-s − 1.87·51-s − 11.7·53-s + 4.39·55-s + 5.27·57-s + 5.50·59-s − 3.91·61-s − 4.06·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.790·5-s − 0.407·9-s − 0.749·11-s + 0.638·13-s + 0.608·15-s + 0.340·17-s − 0.907·19-s − 0.00845·23-s − 0.375·25-s + 1.08·27-s + 1.15·29-s + 0.536·31-s + 0.577·33-s + 0.0320·37-s − 0.491·39-s − 0.156·41-s − 0.182·43-s + 0.321·45-s + 1.08·47-s − 0.262·51-s − 1.62·53-s + 0.592·55-s + 0.698·57-s + 0.717·59-s − 0.501·61-s − 0.504·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 3.95T + 19T^{2} \) |
| 23 | \( 1 + 0.0405T + 23T^{2} \) |
| 29 | \( 1 - 6.20T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 - 0.195T + 37T^{2} \) |
| 43 | \( 1 + 1.19T + 43T^{2} \) |
| 47 | \( 1 - 7.43T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 5.50T + 59T^{2} \) |
| 61 | \( 1 + 3.91T + 61T^{2} \) |
| 67 | \( 1 - 5.79T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 1.35T + 79T^{2} \) |
| 83 | \( 1 + 6.02T + 83T^{2} \) |
| 89 | \( 1 - 0.226T + 89T^{2} \) |
| 97 | \( 1 + 1.43T + 97T^{2} \) |
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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
−7.55447158122746232950466364388, −6.61278701581809932577224595085, −6.17403877039032103126456327356, −5.37675280421962570956792943644, −4.74813956905980877643093646590, −3.97378591381469387052890044729, −3.15960943789900762050316749374, −2.29623122464111008575895203888, −0.946560113574448147551350046232, 0,
0.946560113574448147551350046232, 2.29623122464111008575895203888, 3.15960943789900762050316749374, 3.97378591381469387052890044729, 4.74813956905980877643093646590, 5.37675280421962570956792943644, 6.17403877039032103126456327356, 6.61278701581809932577224595085, 7.55447158122746232950466364388
