| L(s) = 1 | + 1.82·2-s + 1.34·4-s + 3.32·5-s − 0.206·7-s − 1.19·8-s + 6.08·10-s − 5.29·11-s + 6.12·13-s − 0.377·14-s − 4.87·16-s + 1.64·17-s + 4.95·19-s + 4.48·20-s − 9.69·22-s + 1.58·23-s + 6.05·25-s + 11.2·26-s − 0.278·28-s + 7.81·29-s + 5.10·31-s − 6.54·32-s + 3.00·34-s − 0.686·35-s − 2.40·37-s + 9.06·38-s − 3.96·40-s + 8.27·41-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.674·4-s + 1.48·5-s − 0.0780·7-s − 0.421·8-s + 1.92·10-s − 1.59·11-s + 1.69·13-s − 0.100·14-s − 1.21·16-s + 0.398·17-s + 1.13·19-s + 1.00·20-s − 2.06·22-s + 0.330·23-s + 1.21·25-s + 2.19·26-s − 0.0526·28-s + 1.45·29-s + 0.916·31-s − 1.15·32-s + 0.515·34-s − 0.116·35-s − 0.395·37-s + 1.47·38-s − 0.626·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.374156016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.374156016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 5 | \( 1 - 3.32T + 5T^{2} \) |
| 7 | \( 1 + 0.206T + 7T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 - 7.81T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 + 2.40T + 37T^{2} \) |
| 41 | \( 1 - 8.27T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 - 5.76T + 47T^{2} \) |
| 53 | \( 1 + 1.67T + 53T^{2} \) |
| 59 | \( 1 + 9.59T + 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 + 0.444T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 7.20T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 2.00T + 83T^{2} \) |
| 89 | \( 1 + 18.7T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−9.120381360783766552088613635099, −8.362251909869040383789742186585, −7.30812673574590755376001907168, −6.12257558062865306272465838108, −5.92007095101620779831869676384, −5.18591455205962287385447352993, −4.39113281391500047493359026114, −3.08106788807139913478971796387, −2.68762395951365895476484459500, −1.25981455277541141898014519222,
1.25981455277541141898014519222, 2.68762395951365895476484459500, 3.08106788807139913478971796387, 4.39113281391500047493359026114, 5.18591455205962287385447352993, 5.92007095101620779831869676384, 6.12257558062865306272465838108, 7.30812673574590755376001907168, 8.362251909869040383789742186585, 9.120381360783766552088613635099
