| L(s) = 1 | − 2-s − 2.60·3-s + 4-s + 0.943·5-s + 2.60·6-s + 3.39·7-s − 8-s + 3.77·9-s − 0.943·10-s − 4.92·11-s − 2.60·12-s + 5.95·13-s − 3.39·14-s − 2.45·15-s + 16-s + 3.05·17-s − 3.77·18-s + 1.84·19-s + 0.943·20-s − 8.84·21-s + 4.92·22-s − 9.14·23-s + 2.60·24-s − 4.10·25-s − 5.95·26-s − 2.02·27-s + 3.39·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.50·3-s + 0.5·4-s + 0.422·5-s + 1.06·6-s + 1.28·7-s − 0.353·8-s + 1.25·9-s − 0.298·10-s − 1.48·11-s − 0.751·12-s + 1.65·13-s − 0.908·14-s − 0.634·15-s + 0.250·16-s + 0.740·17-s − 0.890·18-s + 0.422·19-s + 0.211·20-s − 1.93·21-s + 1.05·22-s − 1.90·23-s + 0.531·24-s − 0.821·25-s − 1.16·26-s − 0.389·27-s + 0.642·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003380275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003380275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 2.60T + 3T^{2} \) |
| 5 | \( 1 - 0.943T + 5T^{2} \) |
| 7 | \( 1 - 3.39T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 + 9.14T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 0.125T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 + 0.237T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 0.664T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 3.66T + 67T^{2} \) |
| 71 | \( 1 - 6.13T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 - 5.62T + 83T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
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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
−8.004277412626893789273902238403, −7.61936382702326786002341704087, −6.39900809812505577940112491459, −5.97121798220397925429186307240, −5.41556979896471815345663388420, −4.75849434239767550914056589186, −3.76124026396673294710402238849, −2.41768924268253237238763334022, −1.49947826790073991693574916567, −0.68201970547878196082964732756,
0.68201970547878196082964732756, 1.49947826790073991693574916567, 2.41768924268253237238763334022, 3.76124026396673294710402238849, 4.75849434239767550914056589186, 5.41556979896471815345663388420, 5.97121798220397925429186307240, 6.39900809812505577940112491459, 7.61936382702326786002341704087, 8.004277412626893789273902238403
