| L(s) = 1 | + 2.35·2-s + 1.28·3-s + 3.53·4-s − 3.75·5-s + 3.01·6-s + 3.59·8-s − 1.35·9-s − 8.84·10-s + 4.39·11-s + 4.52·12-s + 0.364·13-s − 4.81·15-s + 1.40·16-s + 0.0294·17-s − 3.19·18-s + 8.08·19-s − 13.2·20-s + 10.3·22-s + 6.85·23-s + 4.61·24-s + 9.13·25-s + 0.856·26-s − 5.58·27-s + 7.02·29-s − 11.3·30-s + 0.530·31-s − 3.89·32-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.739·3-s + 1.76·4-s − 1.68·5-s + 1.22·6-s + 1.27·8-s − 0.452·9-s − 2.79·10-s + 1.32·11-s + 1.30·12-s + 0.101·13-s − 1.24·15-s + 0.350·16-s + 0.00715·17-s − 0.753·18-s + 1.85·19-s − 2.96·20-s + 2.20·22-s + 1.42·23-s + 0.941·24-s + 1.82·25-s + 0.167·26-s − 1.07·27-s + 1.30·29-s − 2.06·30-s + 0.0953·31-s − 0.689·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.352343952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.352343952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 - 0.364T + 13T^{2} \) |
| 17 | \( 1 - 0.0294T + 17T^{2} \) |
| 19 | \( 1 - 8.08T + 19T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 - 7.02T + 29T^{2} \) |
| 31 | \( 1 - 0.530T + 31T^{2} \) |
| 37 | \( 1 - 4.77T + 37T^{2} \) |
| 41 | \( 1 + 5.82T + 41T^{2} \) |
| 43 | \( 1 - 0.400T + 43T^{2} \) |
| 47 | \( 1 + 8.75T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 - 6.91T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 - 1.44T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 3.17T + 73T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 3.21T + 89T^{2} \) |
| 97 | \( 1 + 6.80T + 97T^{2} \) |
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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
−8.433321057564754909946451199010, −7.47202153434718177081342452068, −7.02438004676596637078424408768, −6.21938972918743345440702135378, −5.16226327017159762175569638458, −4.57791613457549297228875692923, −3.63692714823477898325917407978, −3.40342999496567407127428215317, −2.66381919628251221877750723197, −1.05683141779855424067916034867,
1.05683141779855424067916034867, 2.66381919628251221877750723197, 3.40342999496567407127428215317, 3.63692714823477898325917407978, 4.57791613457549297228875692923, 5.16226327017159762175569638458, 6.21938972918743345440702135378, 7.02438004676596637078424408768, 7.47202153434718177081342452068, 8.433321057564754909946451199010
