| L(s) = 1 | + 2.23·2-s + 3.02·3-s + 3.00·4-s − 3.28·5-s + 6.76·6-s + 7-s + 2.24·8-s + 6.15·9-s − 7.35·10-s + 3.69·11-s + 9.09·12-s + 2.23·14-s − 9.94·15-s − 0.979·16-s + 0.705·17-s + 13.7·18-s + 0.911·19-s − 9.87·20-s + 3.02·21-s + 8.27·22-s − 3.01·23-s + 6.80·24-s + 5.79·25-s + 9.55·27-s + 3.00·28-s + 6.55·29-s − 22.2·30-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.74·3-s + 1.50·4-s − 1.46·5-s + 2.76·6-s + 0.377·7-s + 0.795·8-s + 2.05·9-s − 2.32·10-s + 1.11·11-s + 2.62·12-s + 0.597·14-s − 2.56·15-s − 0.244·16-s + 0.171·17-s + 3.24·18-s + 0.209·19-s − 2.20·20-s + 0.660·21-s + 1.76·22-s − 0.627·23-s + 1.38·24-s + 1.15·25-s + 1.83·27-s + 0.567·28-s + 1.21·29-s − 4.06·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.832380400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.832380400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 - 3.02T + 3T^{2} \) |
| 5 | \( 1 + 3.28T + 5T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 17 | \( 1 - 0.705T + 17T^{2} \) |
| 19 | \( 1 - 0.911T + 19T^{2} \) |
| 23 | \( 1 + 3.01T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 + 9.55T + 31T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 - 0.884T + 41T^{2} \) |
| 43 | \( 1 - 0.536T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 + 1.81T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 9.72T + 73T^{2} \) |
| 79 | \( 1 + 7.00T + 79T^{2} \) |
| 83 | \( 1 + 2.31T + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 - 8.79T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.563007188271695853827006131425, −8.720113896008199766212910371995, −8.055473679936979089004838390788, −7.28027175753920031122910964851, −6.57305781783063947169492542953, −5.07621909005894190478120802985, −4.12643779423022479297818057811, −3.72608269986068989580643005017, −3.04954903806975503408224351265, −1.79697540812866245580030997876,
1.79697540812866245580030997876, 3.04954903806975503408224351265, 3.72608269986068989580643005017, 4.12643779423022479297818057811, 5.07621909005894190478120802985, 6.57305781783063947169492542953, 7.28027175753920031122910964851, 8.055473679936979089004838390788, 8.720113896008199766212910371995, 9.563007188271695853827006131425
