| L(s) = 1 | + 2.18·3-s + 0.493·5-s − 1.18·7-s + 1.76·9-s − 11-s − 1.80·13-s + 1.07·15-s − 6.21·17-s − 19-s − 2.58·21-s − 4.62·23-s − 4.75·25-s − 2.68·27-s + 4.44·29-s − 0.457·31-s − 2.18·33-s − 0.584·35-s − 3.58·37-s − 3.93·39-s − 1.33·41-s + 2.75·43-s + 0.873·45-s − 3.97·47-s − 5.59·49-s − 13.5·51-s − 6.98·53-s − 0.493·55-s + ⋯ |
| L(s) = 1 | + 1.26·3-s + 0.220·5-s − 0.447·7-s + 0.589·9-s − 0.301·11-s − 0.500·13-s + 0.278·15-s − 1.50·17-s − 0.229·19-s − 0.564·21-s − 0.965·23-s − 0.951·25-s − 0.517·27-s + 0.825·29-s − 0.0822·31-s − 0.380·33-s − 0.0988·35-s − 0.589·37-s − 0.630·39-s − 0.208·41-s + 0.420·43-s + 0.130·45-s − 0.579·47-s − 0.799·49-s − 1.90·51-s − 0.959·53-s − 0.0665·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 - 0.493T + 5T^{2} \) |
| 7 | \( 1 + 1.18T + 7T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 + 6.21T + 17T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 + 0.457T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 - 2.75T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 + 6.98T + 53T^{2} \) |
| 59 | \( 1 + 0.415T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 + 3.28T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 + 0.391T + 73T^{2} \) |
| 79 | \( 1 - 1.14T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 3.06T + 97T^{2} \) |
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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
−8.267559542522712190957252471007, −7.74348157497290494193436151392, −6.78443142466373043398339828488, −6.17365003679573478880966785672, −5.08474661021959087709527718764, −4.18384171204454978971949980825, −3.40088604129924937321526429237, −2.47096660799521095962961951858, −1.93174727282761731305088401382, 0,
1.93174727282761731305088401382, 2.47096660799521095962961951858, 3.40088604129924937321526429237, 4.18384171204454978971949980825, 5.08474661021959087709527718764, 6.17365003679573478880966785672, 6.78443142466373043398339828488, 7.74348157497290494193436151392, 8.267559542522712190957252471007
