| L(s) = 1 | + 1.95·2-s + 3-s + 1.81·4-s + 3.98·5-s + 1.95·6-s + 1.61·7-s − 0.367·8-s + 9-s + 7.77·10-s + 1.81·12-s − 13-s + 3.15·14-s + 3.98·15-s − 4.34·16-s − 5.81·17-s + 1.95·18-s + 4.50·19-s + 7.21·20-s + 1.61·21-s + 4.15·23-s − 0.367·24-s + 10.8·25-s − 1.95·26-s + 27-s + 2.92·28-s + 5.56·29-s + 7.77·30-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.577·3-s + 0.905·4-s + 1.78·5-s + 0.797·6-s + 0.609·7-s − 0.130·8-s + 0.333·9-s + 2.45·10-s + 0.522·12-s − 0.277·13-s + 0.841·14-s + 1.02·15-s − 1.08·16-s − 1.40·17-s + 0.460·18-s + 1.03·19-s + 1.61·20-s + 0.352·21-s + 0.866·23-s − 0.0750·24-s + 2.17·25-s − 0.382·26-s + 0.192·27-s + 0.552·28-s + 1.03·29-s + 1.41·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.374606862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.374606862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.95T + 2T^{2} \) |
| 5 | \( 1 - 3.98T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 17 | \( 1 + 5.81T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 - 5.11T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 - 0.458T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 2.20T + 53T^{2} \) |
| 59 | \( 1 - 4.02T + 59T^{2} \) |
| 61 | \( 1 - 8.20T + 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 2.19T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 8.95T + 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 - 2.71T + 97T^{2} \) |
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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
−8.446891403700137674300118451326, −7.21224271707836253853445227037, −6.67547799212465255351530570826, −5.91427451155032118584371268886, −5.18171232019235464797890792036, −4.80108349274973524511579186105, −3.84301813013204698448999702542, −2.73119245063797898745396039930, −2.37152546215005778426331388088, −1.35634620906172762534059490032,
1.35634620906172762534059490032, 2.37152546215005778426331388088, 2.73119245063797898745396039930, 3.84301813013204698448999702542, 4.80108349274973524511579186105, 5.18171232019235464797890792036, 5.91427451155032118584371268886, 6.67547799212465255351530570826, 7.21224271707836253853445227037, 8.446891403700137674300118451326
