| L(s) = 1 | + (−1.06 − 1.36i)3-s − 2.12i·5-s + (−0.732 + 2.90i)9-s − 5.81·11-s − 4.19·13-s + (−2.90 + 2.26i)15-s + 5.81i·17-s − 2.73i·19-s + 4.25·23-s + 0.464·25-s + (4.75 − 2.09i)27-s + 5.81i·29-s + 2.53i·31-s + (6.19 + 7.94i)33-s + 11.4·37-s + ⋯ |
| L(s) = 1 | + (−0.614 − 0.788i)3-s − 0.952i·5-s + (−0.244 + 0.969i)9-s − 1.75·11-s − 1.16·13-s + (−0.751 + 0.585i)15-s + 1.41i·17-s − 0.626i·19-s + 0.888·23-s + 0.0928·25-s + (0.914 − 0.403i)27-s + 1.08i·29-s + 0.455i·31-s + (1.07 + 1.38i)33-s + 1.88·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7849415764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7849415764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.06 + 1.36i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.12iT - 5T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 - 5.81iT - 17T^{2} \) |
| 19 | \( 1 + 2.73iT - 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 - 5.81iT - 29T^{2} \) |
| 31 | \( 1 - 2.53iT - 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 1.55iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 1.55iT - 53T^{2} \) |
| 59 | \( 1 + 9.50T + 59T^{2} \) |
| 61 | \( 1 - 1.26T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 1.55T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 + 9.50T + 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
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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
−8.819396647448450428473143687741, −8.101015317496853090077475819392, −7.54203045077697111610197075064, −6.73928654355222548457765542272, −5.73151026837944059753802099923, −5.05466400971549134419441404856, −4.62963150939371459539502742366, −2.96448252834274438257801605128, −2.06122536661737339631991856370, −0.833912395286505395716069370992,
0.38593961862057597301312229012, 2.65868215345599597197930084175, 2.89814142546551520013835342707, 4.31591294119446204034900891102, 5.03201651460858212138092326529, 5.68402929340398672511462566667, 6.62184026285290732583720676301, 7.44038028213720766245674786143, 7.991736285835690249700520989542, 9.315868802453981619663607741811
