| L(s) = 1 | − 1.54·2-s − 3.21·3-s + 0.401·4-s − 2.07·5-s + 4.98·6-s + 0.871·7-s + 2.47·8-s + 7.33·9-s + 3.22·10-s + 0.118·11-s − 1.29·12-s − 1.17·13-s − 1.35·14-s + 6.68·15-s − 4.64·16-s + 6.72·17-s − 11.3·18-s + 4.91·19-s − 0.835·20-s − 2.80·21-s − 0.184·22-s + 2.58·23-s − 7.96·24-s − 0.677·25-s + 1.81·26-s − 13.9·27-s + 0.350·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 1.85·3-s + 0.200·4-s − 0.929·5-s + 2.03·6-s + 0.329·7-s + 0.875·8-s + 2.44·9-s + 1.01·10-s + 0.0358·11-s − 0.372·12-s − 0.324·13-s − 0.360·14-s + 1.72·15-s − 1.16·16-s + 1.63·17-s − 2.67·18-s + 1.12·19-s − 0.186·20-s − 0.611·21-s − 0.0392·22-s + 0.539·23-s − 1.62·24-s − 0.135·25-s + 0.356·26-s − 2.68·27-s + 0.0661·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3697423241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3697423241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 3 | \( 1 + 3.21T + 3T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 7 | \( 1 - 0.871T + 7T^{2} \) |
| 11 | \( 1 - 0.118T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 - 4.91T + 19T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 - 0.929T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 0.871T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 47 | \( 1 + 9.72T + 47T^{2} \) |
| 53 | \( 1 + 5.24T + 53T^{2} \) |
| 59 | \( 1 + 3.87T + 59T^{2} \) |
| 61 | \( 1 + 2.03T + 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 3.30T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 + 2.34T + 83T^{2} \) |
| 89 | \( 1 - 7.20T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−9.706391092208799677301589770721, −8.230324502595439407104885376810, −7.68873969630221335188914465492, −7.09507295564395779401345803843, −6.12522986208866481412288565652, −5.07271510986688262614721387440, −4.69225133902853288375928887700, −3.49955788533836509871842451823, −1.39908032929780423650383653962, −0.60833321310490738539897442199,
0.60833321310490738539897442199, 1.39908032929780423650383653962, 3.49955788533836509871842451823, 4.69225133902853288375928887700, 5.07271510986688262614721387440, 6.12522986208866481412288565652, 7.09507295564395779401345803843, 7.68873969630221335188914465492, 8.230324502595439407104885376810, 9.706391092208799677301589770721
