| L(s) = 1 | − 2.45·3-s + 5-s + 1.57·7-s + 3.00·9-s + 2.43·11-s + 5.07·13-s − 2.45·15-s + 2.43·17-s − 3.24·19-s − 3.86·21-s + 0.280·23-s + 25-s − 0.0109·27-s + 10.4·29-s + 1.50·31-s − 5.97·33-s + 1.57·35-s − 37-s − 12.4·39-s + 3.14·41-s − 2.51·43-s + 3.00·45-s − 8.21·47-s − 4.51·49-s − 5.96·51-s − 5.33·53-s + 2.43·55-s + ⋯ |
| L(s) = 1 | − 1.41·3-s + 0.447·5-s + 0.596·7-s + 1.00·9-s + 0.734·11-s + 1.40·13-s − 0.632·15-s + 0.589·17-s − 0.743·19-s − 0.843·21-s + 0.0585·23-s + 0.200·25-s − 0.00210·27-s + 1.93·29-s + 0.269·31-s − 1.03·33-s + 0.266·35-s − 0.164·37-s − 1.99·39-s + 0.490·41-s − 0.383·43-s + 0.447·45-s − 1.19·47-s − 0.644·49-s − 0.834·51-s − 0.733·53-s + 0.328·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516822081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516822081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 2.45T + 3T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 - 5.07T + 13T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 + 3.24T + 19T^{2} \) |
| 23 | \( 1 - 0.280T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 41 | \( 1 - 3.14T + 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 + 5.33T + 53T^{2} \) |
| 59 | \( 1 + 3.43T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 - 1.94T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 73 | \( 1 + 8.31T + 73T^{2} \) |
| 79 | \( 1 - 6.72T + 79T^{2} \) |
| 83 | \( 1 + 0.545T + 83T^{2} \) |
| 89 | \( 1 + 7.33T + 89T^{2} \) |
| 97 | \( 1 + 9.99T + 97T^{2} \) |
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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
−8.598903606501672697434976489252, −8.155093987374733688358686507578, −6.84035283343085258978536036118, −6.36391493202637278557273799185, −5.82043773428503300278355288407, −4.94121087876681801882233430746, −4.31288964380011271741186756774, −3.17831551313197178504953040772, −1.66901482396835802437432880622, −0.885847321108398293540487704017,
0.885847321108398293540487704017, 1.66901482396835802437432880622, 3.17831551313197178504953040772, 4.31288964380011271741186756774, 4.94121087876681801882233430746, 5.82043773428503300278355288407, 6.36391493202637278557273799185, 6.84035283343085258978536036118, 8.155093987374733688358686507578, 8.598903606501672697434976489252
