| L(s) = 1 | + 2.41·2-s + 3.82·4-s − 2·5-s + 4.41·8-s − 4.82·10-s − 11-s + 0.828·13-s + 2.99·16-s − 4.41·17-s − 7.24·19-s − 7.65·20-s − 2.41·22-s + 7·23-s − 25-s + 1.99·26-s − 3.24·29-s − 5.65·31-s − 1.58·32-s − 10.6·34-s − 9.48·37-s − 17.4·38-s − 8.82·40-s − 1.17·41-s − 2.75·43-s − 3.82·44-s + 16.8·46-s + 9.82·47-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s − 0.894·5-s + 1.56·8-s − 1.52·10-s − 0.301·11-s + 0.229·13-s + 0.749·16-s − 1.07·17-s − 1.66·19-s − 1.71·20-s − 0.514·22-s + 1.45·23-s − 0.200·25-s + 0.392·26-s − 0.602·29-s − 1.01·31-s − 0.280·32-s − 1.82·34-s − 1.55·37-s − 2.83·38-s − 1.39·40-s − 0.182·41-s − 0.420·43-s − 0.577·44-s + 2.49·46-s + 1.43·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 - 7T + 23T^{2} \) |
| 29 | \( 1 + 3.24T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 9.48T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 + 2.75T + 43T^{2} \) |
| 47 | \( 1 - 9.82T + 47T^{2} \) |
| 53 | \( 1 - 7.17T + 53T^{2} \) |
| 59 | \( 1 + 8.65T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 - 4.17T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.57085489830467025490857425681, −6.96315287568312212206374229403, −6.40352154193492582690094313220, −5.49208405621410475664084915989, −4.87376010719391158869986270644, −4.06552336825994382101818226662, −3.68996633900787733164742912888, −2.69160792032119484006769477906, −1.87506268527371296560263131990, 0,
1.87506268527371296560263131990, 2.69160792032119484006769477906, 3.68996633900787733164742912888, 4.06552336825994382101818226662, 4.87376010719391158869986270644, 5.49208405621410475664084915989, 6.40352154193492582690094313220, 6.96315287568312212206374229403, 7.57085489830467025490857425681
