| L(s) = 1 | − 2.62·2-s + 4.88·4-s − 0.357·5-s − 2.31·7-s − 7.58·8-s + 0.937·10-s − 4.12·11-s + 5.02·13-s + 6.06·14-s + 10.1·16-s − 5.44·17-s + 19-s − 1.74·20-s + 10.8·22-s − 3.18·23-s − 4.87·25-s − 13.1·26-s − 11.3·28-s − 0.473·29-s + 0.270·31-s − 11.3·32-s + 14.2·34-s + 0.826·35-s − 9.94·37-s − 2.62·38-s + 2.70·40-s + 6.51·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.44·4-s − 0.159·5-s − 0.874·7-s − 2.68·8-s + 0.296·10-s − 1.24·11-s + 1.39·13-s + 1.62·14-s + 2.52·16-s − 1.32·17-s + 0.229·19-s − 0.390·20-s + 2.30·22-s − 0.664·23-s − 0.974·25-s − 2.58·26-s − 2.13·28-s − 0.0879·29-s + 0.0486·31-s − 2.01·32-s + 2.45·34-s + 0.139·35-s − 1.63·37-s − 0.425·38-s + 0.428·40-s + 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2276145026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2276145026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 + 0.357T + 5T^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 + 0.473T + 29T^{2} \) |
| 31 | \( 1 - 0.270T + 31T^{2} \) |
| 37 | \( 1 + 9.94T + 37T^{2} \) |
| 41 | \( 1 - 6.51T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 6.04T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 8.02T + 73T^{2} \) |
| 79 | \( 1 + 4.73T + 79T^{2} \) |
| 83 | \( 1 - 2.57T + 83T^{2} \) |
| 89 | \( 1 + 4.01T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.045186967401195215391288376354, −7.33381729860297800042579929224, −6.64220315190216373840700279684, −6.14592021474008549679590859179, −5.36484068174926563907653366036, −4.02221762402976504396006240306, −3.19073843234372303957564176936, −2.37085330313311324903699275747, −1.57857008876908657988172742457, −0.30583720867504754487265438062,
0.30583720867504754487265438062, 1.57857008876908657988172742457, 2.37085330313311324903699275747, 3.19073843234372303957564176936, 4.02221762402976504396006240306, 5.36484068174926563907653366036, 6.14592021474008549679590859179, 6.64220315190216373840700279684, 7.33381729860297800042579929224, 8.045186967401195215391288376354
