| L(s) = 1 | − 5-s + 1.26·7-s + 5.46·11-s + 4·13-s + 4·17-s + 6.19·19-s + 8.92·23-s + 25-s − 7.46·29-s + 10.1·31-s − 1.26·35-s + 37-s − 4.92·41-s + 6·43-s − 1.26·47-s − 5.39·49-s − 12.9·53-s − 5.46·55-s − 3.66·59-s + 14·61-s − 4·65-s − 5.26·67-s − 8·71-s − 10.3·73-s + 6.92·77-s + 6.19·79-s − 1.66·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.479·7-s + 1.64·11-s + 1.10·13-s + 0.970·17-s + 1.42·19-s + 1.86·23-s + 0.200·25-s − 1.38·29-s + 1.83·31-s − 0.214·35-s + 0.164·37-s − 0.769·41-s + 0.914·43-s − 0.184·47-s − 0.770·49-s − 1.77·53-s − 0.736·55-s − 0.476·59-s + 1.79·61-s − 0.496·65-s − 0.643·67-s − 0.949·71-s − 1.21·73-s + 0.789·77-s + 0.697·79-s − 0.182·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.850563865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.850563865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 3.66T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 5.26T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 6.19T + 79T^{2} \) |
| 83 | \( 1 + 1.66T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−7.985796035268412574264512571425, −7.30003461503997691114460293543, −6.64194599837109156851624099607, −5.90989875910007022026534618012, −5.11063074685569888081893924172, −4.34665405293149061196932876670, −3.51683620110618350610943882838, −3.04029320729344446829243626781, −1.40885739769473434892341782349, −1.06571257106687043127735882754,
1.06571257106687043127735882754, 1.40885739769473434892341782349, 3.04029320729344446829243626781, 3.51683620110618350610943882838, 4.34665405293149061196932876670, 5.11063074685569888081893924172, 5.90989875910007022026534618012, 6.64194599837109156851624099607, 7.30003461503997691114460293543, 7.985796035268412574264512571425
