| L(s) = 1 | + 1.30·3-s + 5-s + 2.73·7-s − 1.29·9-s + 5.91·11-s + 1.05·13-s + 1.30·15-s − 0.377·17-s + 19-s + 3.57·21-s + 2.73·23-s + 25-s − 5.60·27-s − 7.41·29-s + 5.83·31-s + 7.72·33-s + 2.73·35-s + 2.23·37-s + 1.37·39-s + 12.5·41-s − 6.67·43-s − 1.29·45-s − 4.95·47-s + 0.507·49-s − 0.492·51-s − 0.187·53-s + 5.91·55-s + ⋯ |
| L(s) = 1 | + 0.753·3-s + 0.447·5-s + 1.03·7-s − 0.432·9-s + 1.78·11-s + 0.293·13-s + 0.336·15-s − 0.0915·17-s + 0.229·19-s + 0.780·21-s + 0.571·23-s + 0.200·25-s − 1.07·27-s − 1.37·29-s + 1.04·31-s + 1.34·33-s + 0.463·35-s + 0.367·37-s + 0.220·39-s + 1.96·41-s − 1.01·43-s − 0.193·45-s − 0.722·47-s + 0.0724·49-s − 0.0689·51-s − 0.0257·53-s + 0.798·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.304444731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.304444731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 + 0.377T + 17T^{2} \) |
| 23 | \( 1 - 2.73T + 23T^{2} \) |
| 29 | \( 1 + 7.41T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 - 2.23T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 6.67T + 43T^{2} \) |
| 47 | \( 1 + 4.95T + 47T^{2} \) |
| 53 | \( 1 + 0.187T + 53T^{2} \) |
| 59 | \( 1 + 6.10T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 9.60T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.811875723572380481716577929263, −8.087337421506021440712576206010, −7.35441314797049900491406577257, −6.40885912129126566330203757871, −5.75255349086693175596167188767, −4.74076896930831047465463388832, −3.94503733549053017719139834286, −3.06994597552166897076753517606, −1.99692080057928023915629512605, −1.20126548886843213335841646053,
1.20126548886843213335841646053, 1.99692080057928023915629512605, 3.06994597552166897076753517606, 3.94503733549053017719139834286, 4.74076896930831047465463388832, 5.75255349086693175596167188767, 6.40885912129126566330203757871, 7.35441314797049900491406577257, 8.087337421506021440712576206010, 8.811875723572380481716577929263
