| L(s) = 1 | + 0.0931·2-s − 1.99·4-s + 5-s + 1.36·7-s − 0.371·8-s + 0.0931·10-s − 11-s − 1.08·13-s + 0.127·14-s + 3.94·16-s − 2.21·17-s + 19-s − 1.99·20-s − 0.0931·22-s − 3.71·23-s + 25-s − 0.101·26-s − 2.71·28-s + 3.62·29-s − 1.08·31-s + 1.11·32-s − 0.206·34-s + 1.36·35-s − 7.37·37-s + 0.0931·38-s − 0.371·40-s + 4.24·41-s + ⋯ |
| L(s) = 1 | + 0.0658·2-s − 0.995·4-s + 0.447·5-s + 0.516·7-s − 0.131·8-s + 0.0294·10-s − 0.301·11-s − 0.302·13-s + 0.0339·14-s + 0.987·16-s − 0.537·17-s + 0.229·19-s − 0.445·20-s − 0.0198·22-s − 0.775·23-s + 0.200·25-s − 0.0198·26-s − 0.513·28-s + 0.673·29-s − 0.195·31-s + 0.196·32-s − 0.0353·34-s + 0.230·35-s − 1.21·37-s + 0.0151·38-s − 0.0587·40-s + 0.662·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533526940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533526940\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0931T + 2T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 13 | \( 1 + 1.08T + 13T^{2} \) |
| 17 | \( 1 + 2.21T + 17T^{2} \) |
| 23 | \( 1 + 3.71T + 23T^{2} \) |
| 29 | \( 1 - 3.62T + 29T^{2} \) |
| 31 | \( 1 + 1.08T + 31T^{2} \) |
| 37 | \( 1 + 7.37T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 + 0.232T + 47T^{2} \) |
| 53 | \( 1 - 6.77T + 53T^{2} \) |
| 59 | \( 1 + 4.95T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 8.80T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 + 1.82T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.86741471424216329480236124449, −7.04014717650353089103664615527, −6.25035068191679692816843292676, −5.44836951938901393115070270356, −5.00770057037826946339368133112, −4.29418384349948023200903306442, −3.57591846087369007711822757755, −2.58030241246948764507390330574, −1.71684226752544131847442759048, −0.59614268806840270140597331334,
0.59614268806840270140597331334, 1.71684226752544131847442759048, 2.58030241246948764507390330574, 3.57591846087369007711822757755, 4.29418384349948023200903306442, 5.00770057037826946339368133112, 5.44836951938901393115070270356, 6.25035068191679692816843292676, 7.04014717650353089103664615527, 7.86741471424216329480236124449
