| L(s) = 1 | − 0.668·2-s − 1.31·3-s − 1.55·4-s − 3.52·5-s + 0.877·6-s + 2.37·8-s − 1.27·9-s + 2.35·10-s − 2.97·11-s + 2.03·12-s + 4.03·13-s + 4.62·15-s + 1.51·16-s + 3.69·17-s + 0.852·18-s + 8.19·19-s + 5.47·20-s + 1.98·22-s − 1.70·23-s − 3.11·24-s + 7.43·25-s − 2.69·26-s + 5.61·27-s − 4.73·29-s − 3.09·30-s + 31-s − 5.76·32-s + ⋯ |
| L(s) = 1 | − 0.472·2-s − 0.758·3-s − 0.776·4-s − 1.57·5-s + 0.358·6-s + 0.839·8-s − 0.425·9-s + 0.745·10-s − 0.896·11-s + 0.588·12-s + 1.11·13-s + 1.19·15-s + 0.379·16-s + 0.896·17-s + 0.201·18-s + 1.87·19-s + 1.22·20-s + 0.423·22-s − 0.354·23-s − 0.636·24-s + 1.48·25-s − 0.529·26-s + 1.08·27-s − 0.880·29-s − 0.565·30-s + 0.179·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.668T + 2T^{2} \) |
| 3 | \( 1 + 1.31T + 3T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 11 | \( 1 + 2.97T + 11T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 - 8.19T + 19T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 + 4.69T + 43T^{2} \) |
| 47 | \( 1 + 9.96T + 47T^{2} \) |
| 53 | \( 1 - 9.59T + 53T^{2} \) |
| 59 | \( 1 + 6.50T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 6.68T + 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 + 1.60T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 4.81T + 97T^{2} \) |
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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
−8.927106966366031480211571711023, −8.063697713094732670592406006530, −7.85595817802926371270336013757, −6.81204912643637966180192453412, −5.41123020435010513224111979526, −5.14950689766829830531863532054, −3.83092448253156736101930937732, −3.28684000523049100375376356315, −1.04744916116378114429027685448, 0,
1.04744916116378114429027685448, 3.28684000523049100375376356315, 3.83092448253156736101930937732, 5.14950689766829830531863532054, 5.41123020435010513224111979526, 6.81204912643637966180192453412, 7.85595817802926371270336013757, 8.063697713094732670592406006530, 8.927106966366031480211571711023
