| L(s) = 1 | − 3-s + 3.46·7-s − 2·9-s + 3·11-s + 3.46·13-s − 3·17-s − 19-s − 3.46·21-s + 5·27-s − 10.3·29-s − 6.92·31-s − 3·33-s − 10.3·37-s − 3.46·39-s + 9·41-s − 4·43-s − 10.3·47-s + 4.99·49-s + 3·51-s + 57-s − 12·59-s + 3.46·61-s − 6.92·63-s − 11·67-s + 10.3·71-s + 7·73-s + 10.3·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.30·7-s − 0.666·9-s + 0.904·11-s + 0.960·13-s − 0.727·17-s − 0.229·19-s − 0.755·21-s + 0.962·27-s − 1.92·29-s − 1.24·31-s − 0.522·33-s − 1.70·37-s − 0.554·39-s + 1.40·41-s − 0.609·43-s − 1.51·47-s + 0.714·49-s + 0.420·51-s + 0.132·57-s − 1.56·59-s + 0.443·61-s − 0.872·63-s − 1.34·67-s + 1.23·71-s + 0.819·73-s + 1.18·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 + 11T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−7.70335810156829705863391904265, −6.89196383428079211455618868444, −6.18547764499815853950078549552, −5.52931567541923221027731900029, −4.91517786982457014928812181189, −4.05831087742214772749981375137, −3.37140572117864993375075955325, −2.00416229582808273777504770876, −1.41508994746583727573027361004, 0,
1.41508994746583727573027361004, 2.00416229582808273777504770876, 3.37140572117864993375075955325, 4.05831087742214772749981375137, 4.91517786982457014928812181189, 5.52931567541923221027731900029, 6.18547764499815853950078549552, 6.89196383428079211455618868444, 7.70335810156829705863391904265
