| L(s) = 1 | − 5-s − 5.24·7-s − 2.67·11-s + 3.81·13-s − 3.52·17-s − 4.67·19-s + 4.95·23-s + 25-s + 1.85·29-s − 8.67·31-s + 5.24·35-s − 2.67·37-s + 3.67·41-s + 3.52·43-s − 9.26·47-s + 20.5·49-s + 2.85·53-s + 2.67·55-s − 4.20·59-s − 7.96·61-s − 3.81·65-s − 0.859·67-s + 15.1·71-s + 6.28·73-s + 14.0·77-s + 5.63·79-s + 3.89·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.98·7-s − 0.805·11-s + 1.05·13-s − 0.855·17-s − 1.07·19-s + 1.03·23-s + 0.200·25-s + 0.344·29-s − 1.55·31-s + 0.886·35-s − 0.439·37-s + 0.573·41-s + 0.538·43-s − 1.35·47-s + 2.92·49-s + 0.392·53-s + 0.360·55-s − 0.547·59-s − 1.01·61-s − 0.473·65-s − 0.105·67-s + 1.79·71-s + 0.735·73-s + 1.59·77-s + 0.633·79-s + 0.427·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7752864860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7752864860\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + 5.24T + 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 + 8.67T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 - 3.67T + 41T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 + 9.26T + 47T^{2} \) |
| 53 | \( 1 - 2.85T + 53T^{2} \) |
| 59 | \( 1 + 4.20T + 59T^{2} \) |
| 61 | \( 1 + 7.96T + 61T^{2} \) |
| 67 | \( 1 + 0.859T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 6.28T + 73T^{2} \) |
| 79 | \( 1 - 5.63T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 - 11T + 89T^{2} \) |
| 97 | \( 1 + 3.83T + 97T^{2} \) |
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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
−8.887171853491918548323448686429, −7.903864958291535877307434062300, −6.99229446313867389525283407501, −6.46999931018072146904640354411, −5.81984726395999970787916651233, −4.74840660525938939686854171622, −3.72644720504476567311108355054, −3.22451169527086838302076796577, −2.22183597299868185875472081513, −0.49751083740021633078767928261,
0.49751083740021633078767928261, 2.22183597299868185875472081513, 3.22451169527086838302076796577, 3.72644720504476567311108355054, 4.74840660525938939686854171622, 5.81984726395999970787916651233, 6.46999931018072146904640354411, 6.99229446313867389525283407501, 7.903864958291535877307434062300, 8.887171853491918548323448686429
