| L(s) = 1 | + 0.0559·3-s − 5-s − 0.736·7-s − 2.99·9-s − 1.01·11-s + 5.70·13-s − 0.0559·15-s + 6.01·17-s + 0.103·19-s − 0.0412·21-s − 1.19·23-s + 25-s − 0.335·27-s + 8.14·29-s − 7.50·31-s − 0.0568·33-s + 0.736·35-s + 3.01·37-s + 0.319·39-s − 8.80·41-s + 3.83·43-s + 2.99·45-s − 3.92·47-s − 6.45·49-s + 0.336·51-s + 7.18·53-s + 1.01·55-s + ⋯ |
| L(s) = 1 | + 0.0322·3-s − 0.447·5-s − 0.278·7-s − 0.998·9-s − 0.306·11-s + 1.58·13-s − 0.0144·15-s + 1.45·17-s + 0.0238·19-s − 0.00899·21-s − 0.248·23-s + 0.200·25-s − 0.0645·27-s + 1.51·29-s − 1.34·31-s − 0.00989·33-s + 0.124·35-s + 0.495·37-s + 0.0511·39-s − 1.37·41-s + 0.584·43-s + 0.446·45-s − 0.571·47-s − 0.922·49-s + 0.0471·51-s + 0.986·53-s + 0.137·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.614295505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614295505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0559T + 3T^{2} \) |
| 7 | \( 1 + 0.736T + 7T^{2} \) |
| 11 | \( 1 + 1.01T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 - 6.01T + 17T^{2} \) |
| 19 | \( 1 - 0.103T + 19T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 + 7.50T + 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 + 8.80T + 41T^{2} \) |
| 43 | \( 1 - 3.83T + 43T^{2} \) |
| 47 | \( 1 + 3.92T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 + 3.48T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 + 2.08T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 2.46T + 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 - 1.71T + 83T^{2} \) |
| 89 | \( 1 + 2.35T + 89T^{2} \) |
| 97 | \( 1 + 1.04T + 97T^{2} \) |
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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
−7.978261719415062974976233373132, −7.20172474308241766979045010653, −6.31980439981435209231283949718, −5.80987053052678141966720846816, −5.16439296679020474866707287579, −4.16690742010503896171273742618, −3.32088247725135198047179340475, −3.01413808272444804145361401611, −1.67893134725069473865453947595, −0.63880818248079161513650294570,
0.63880818248079161513650294570, 1.67893134725069473865453947595, 3.01413808272444804145361401611, 3.32088247725135198047179340475, 4.16690742010503896171273742618, 5.16439296679020474866707287579, 5.80987053052678141966720846816, 6.31980439981435209231283949718, 7.20172474308241766979045010653, 7.978261719415062974976233373132
