| L(s) = 1 | + 1.43·3-s − 1.63·5-s − 0.942·9-s + 3.97·11-s + 2.47·13-s − 2.34·15-s − 4.38·17-s + 0.0365·19-s + 0.565·23-s − 2.32·25-s − 5.65·27-s + 2.34·29-s − 0.365·31-s + 5.70·33-s − 1.76·37-s + 3.54·39-s − 41-s − 2.38·43-s + 1.54·45-s − 13.3·47-s − 6.28·51-s − 1.63·53-s − 6.50·55-s + 0.0524·57-s + 5.70·59-s − 12.6·61-s − 4.03·65-s + ⋯ |
| L(s) = 1 | + 0.828·3-s − 0.730·5-s − 0.314·9-s + 1.19·11-s + 0.685·13-s − 0.605·15-s − 1.06·17-s + 0.00838·19-s + 0.117·23-s − 0.465·25-s − 1.08·27-s + 0.435·29-s − 0.0656·31-s + 0.993·33-s − 0.289·37-s + 0.567·39-s − 0.156·41-s − 0.363·43-s + 0.229·45-s − 1.94·47-s − 0.879·51-s − 0.224·53-s − 0.876·55-s + 0.00694·57-s + 0.743·59-s − 1.61·61-s − 0.500·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.43T + 3T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 11 | \( 1 - 3.97T + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 - 0.0365T + 19T^{2} \) |
| 23 | \( 1 - 0.565T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 31 | \( 1 + 0.365T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 1.63T + 53T^{2} \) |
| 59 | \( 1 - 5.70T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 7.10T + 71T^{2} \) |
| 73 | \( 1 - 5.24T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 + 6.30T + 97T^{2} \) |
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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
−7.66853549066086601683556973693, −6.70552415153284390059766797512, −6.38529708729045583798566780701, −5.34386706918352481359010859980, −4.41494547825442648338172826952, −3.76908677275475979357526998035, −3.27964567745329965784525544889, −2.28399173153454719761501409978, −1.38023755789492698481249041305, 0,
1.38023755789492698481249041305, 2.28399173153454719761501409978, 3.27964567745329965784525544889, 3.76908677275475979357526998035, 4.41494547825442648338172826952, 5.34386706918352481359010859980, 6.38529708729045583798566780701, 6.70552415153284390059766797512, 7.66853549066086601683556973693
