| L(s) = 1 | − 2·4-s + 3.86·7-s + 7.20·13-s + 4·16-s − 7.92·19-s − 5·25-s − 7.72·28-s − 9.34·31-s − 8.34·37-s − 3.08·43-s + 7.92·49-s − 14.4·52-s − 15.6·61-s − 8·64-s − 8.13·67-s + 16.2·73-s + 15.8·76-s − 7.51·79-s + 27.8·91-s + 3.18·97-s + 10·100-s − 18.2·103-s − 6.62·109-s + 15.4·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.46·7-s + 1.99·13-s + 16-s − 1.81·19-s − 25-s − 1.46·28-s − 1.67·31-s − 1.37·37-s − 0.469·43-s + 1.13·49-s − 1.99·52-s − 1.99·61-s − 64-s − 0.993·67-s + 1.90·73-s + 1.81·76-s − 0.845·79-s + 2.91·91-s + 0.323·97-s + 100-s − 1.79·103-s − 0.634·109-s + 1.46·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7.20T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 31 | \( 1 + 9.34T + 31T^{2} \) |
| 37 | \( 1 + 8.34T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.6T + 61T^{2} \) |
| 67 | \( 1 + 8.13T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 7.51T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 3.18T + 97T^{2} \) |
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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
−7.931706158776657953257937652113, −6.81846029250075444028909725520, −5.94264702094589428101017023516, −5.44478910106616633733563200060, −4.58560557516145415607200296221, −4.02758453305681070578143329897, −3.44127997653737385813060965211, −1.91164743924746212893676156274, −1.39912475947805497258548745717, 0,
1.39912475947805497258548745717, 1.91164743924746212893676156274, 3.44127997653737385813060965211, 4.02758453305681070578143329897, 4.58560557516145415607200296221, 5.44478910106616633733563200060, 5.94264702094589428101017023516, 6.81846029250075444028909725520, 7.931706158776657953257937652113
