| L(s) = 1 | − 2.06·2-s + 2.11·3-s + 2.27·4-s − 2.43·5-s − 4.38·6-s − 7-s − 0.566·8-s + 1.48·9-s + 5.04·10-s + 3.64·11-s + 4.81·12-s + 2.06·14-s − 5.16·15-s − 3.37·16-s + 7.04·17-s − 3.07·18-s + 2.76·19-s − 5.54·20-s − 2.11·21-s − 7.54·22-s − 7.75·23-s − 1.20·24-s + 0.950·25-s − 3.20·27-s − 2.27·28-s + 2.31·29-s + 10.6·30-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.22·3-s + 1.13·4-s − 1.09·5-s − 1.78·6-s − 0.377·7-s − 0.200·8-s + 0.496·9-s + 1.59·10-s + 1.09·11-s + 1.39·12-s + 0.552·14-s − 1.33·15-s − 0.844·16-s + 1.70·17-s − 0.725·18-s + 0.634·19-s − 1.24·20-s − 0.462·21-s − 1.60·22-s − 1.61·23-s − 0.245·24-s + 0.190·25-s − 0.615·27-s − 0.429·28-s + 0.430·29-s + 1.95·30-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
| \(L(1)\) |
\(\approx\) |
\(0.9577925272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9577925272\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.06T + 2T^{2} \) |
| 3 | \( 1 - 2.11T + 3T^{2} \) |
| 5 | \( 1 + 2.43T + 5T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 17 | \( 1 - 7.04T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 + 7.75T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 7.00T + 31T^{2} \) |
| 37 | \( 1 - 7.28T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 - 3.44T + 43T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 - 9.74T + 71T^{2} \) |
| 73 | \( 1 - 3.75T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 5.42T + 83T^{2} \) |
| 89 | \( 1 - 0.335T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.740461969808188619649425500078, −8.843125384084447481269472390731, −8.211957452596641736474518143730, −7.66863594239903527629371996687, −7.08826185807227946399633319658, −5.79001656835699165300454870172, −4.02983749274013407273846230774, −3.53178778923721581392767103653, −2.23797885529951657600401200401, −0.878651130889336164395992841071,
0.878651130889336164395992841071, 2.23797885529951657600401200401, 3.53178778923721581392767103653, 4.02983749274013407273846230774, 5.79001656835699165300454870172, 7.08826185807227946399633319658, 7.66863594239903527629371996687, 8.211957452596641736474518143730, 8.843125384084447481269472390731, 9.740461969808188619649425500078
