| L(s) = 1 | + 1.35·5-s − 2.73·7-s − 11-s + 5.52·13-s + 4.79·17-s + 3.52·19-s − 0.351·23-s − 3.17·25-s − 3.61·29-s − 7.69·31-s − 3.69·35-s + 7.82·37-s + 0.561·41-s − 5.38·43-s + 5.17·47-s + 0.475·49-s + 8.87·53-s − 1.35·55-s + 5.11·59-s + 4.99·61-s + 7.46·65-s + 7.69·67-s + 12.8·71-s + 4.87·73-s + 2.73·77-s − 5.20·79-s + 12.2·83-s + ⋯ |
| L(s) = 1 | + 0.604·5-s − 1.03·7-s − 0.301·11-s + 1.53·13-s + 1.16·17-s + 0.808·19-s − 0.0733·23-s − 0.634·25-s − 0.670·29-s − 1.38·31-s − 0.624·35-s + 1.28·37-s + 0.0877·41-s − 0.820·43-s + 0.754·47-s + 0.0679·49-s + 1.21·53-s − 0.182·55-s + 0.666·59-s + 0.639·61-s + 0.926·65-s + 0.940·67-s + 1.52·71-s + 0.570·73-s + 0.311·77-s − 0.586·79-s + 1.34·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.910014949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910014949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 1.35T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 13 | \( 1 - 5.52T + 13T^{2} \) |
| 17 | \( 1 - 4.79T + 17T^{2} \) |
| 19 | \( 1 - 3.52T + 19T^{2} \) |
| 23 | \( 1 + 0.351T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 7.69T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 - 0.561T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 - 8.87T + 53T^{2} \) |
| 59 | \( 1 - 5.11T + 59T^{2} \) |
| 61 | \( 1 - 4.99T + 61T^{2} \) |
| 67 | \( 1 - 7.69T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 + 5.20T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−9.161255545329671649058633320569, −8.201933247291578567159002714943, −7.44182968715486463624783571270, −6.52471724489796139912915816449, −5.79993577515767739659360182718, −5.35400836279207393231086022538, −3.80739852861789725542255588449, −3.37906022137469022199065834187, −2.15642629217040240916928780922, −0.911097251257613769412174644799,
0.911097251257613769412174644799, 2.15642629217040240916928780922, 3.37906022137469022199065834187, 3.80739852861789725542255588449, 5.35400836279207393231086022538, 5.79993577515767739659360182718, 6.52471724489796139912915816449, 7.44182968715486463624783571270, 8.201933247291578567159002714943, 9.161255545329671649058633320569
