| L(s) = 1 | + 2.29·3-s + 4.01·5-s + 4.07·7-s + 2.28·9-s − 1.78·11-s − 1.70·13-s + 9.22·15-s + 17-s + 5.98·19-s + 9.36·21-s − 7.29·23-s + 11.1·25-s − 1.65·27-s − 0.931·29-s − 3.34·31-s − 4.09·33-s + 16.3·35-s − 11.5·37-s − 3.91·39-s + 6.51·41-s + 1.48·43-s + 9.15·45-s + 8.42·47-s + 9.61·49-s + 2.29·51-s − 13.6·53-s − 7.15·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 1.79·5-s + 1.54·7-s + 0.760·9-s − 0.537·11-s − 0.472·13-s + 2.38·15-s + 0.242·17-s + 1.37·19-s + 2.04·21-s − 1.52·23-s + 2.22·25-s − 0.317·27-s − 0.172·29-s − 0.601·31-s − 0.712·33-s + 2.76·35-s − 1.89·37-s − 0.627·39-s + 1.01·41-s + 0.225·43-s + 1.36·45-s + 1.22·47-s + 1.37·49-s + 0.321·51-s − 1.87·53-s − 0.964·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.045008471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.045008471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 - 4.01T + 5T^{2} \) |
| 7 | \( 1 - 4.07T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 19 | \( 1 - 5.98T + 19T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 29 | \( 1 + 0.931T + 29T^{2} \) |
| 31 | \( 1 + 3.34T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 6.51T + 41T^{2} \) |
| 43 | \( 1 - 1.48T + 43T^{2} \) |
| 47 | \( 1 - 8.42T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 1.51T + 71T^{2} \) |
| 73 | \( 1 - 1.78T + 73T^{2} \) |
| 79 | \( 1 + 6.46T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 3.18T + 97T^{2} \) |
| 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
−8.532174327151017190702114977226, −7.71719917733798992794917060272, −7.36697889439765483956395609712, −6.06208799354150823584625559932, −5.39254954422221679700777090400, −4.86412603209440959565991575116, −3.67233863805582915939683882006, −2.64137856345318370540539278615, −2.04475927500959291053647391319, −1.44452566658267092019555358080,
1.44452566658267092019555358080, 2.04475927500959291053647391319, 2.64137856345318370540539278615, 3.67233863805582915939683882006, 4.86412603209440959565991575116, 5.39254954422221679700777090400, 6.06208799354150823584625559932, 7.36697889439765483956395609712, 7.71719917733798992794917060272, 8.532174327151017190702114977226
