| L(s) = 1 | − 3.73·5-s + 3.46·7-s + 5.46·11-s + 0.267·13-s + 19-s − 7.46·23-s + 8.92·25-s − 7.46·29-s − 7.46·31-s − 12.9·35-s − 2.92·37-s + 5·41-s − 43-s − 6.66·47-s + 4.99·49-s − 2·53-s − 20.3·55-s + 3.92·59-s − 0.535·61-s − 0.999·65-s + 12.9·67-s − 13.4·71-s − 0.928·73-s + 18.9·77-s − 3.73·79-s + 10.3·83-s − 16.8·89-s + ⋯ |
| L(s) = 1 | − 1.66·5-s + 1.30·7-s + 1.64·11-s + 0.0743·13-s + 0.229·19-s − 1.55·23-s + 1.78·25-s − 1.38·29-s − 1.34·31-s − 2.18·35-s − 0.481·37-s + 0.780·41-s − 0.152·43-s − 0.971·47-s + 0.714·49-s − 0.274·53-s − 2.74·55-s + 0.511·59-s − 0.0686·61-s − 0.124·65-s + 1.57·67-s − 1.59·71-s − 0.108·73-s + 2.15·77-s − 0.419·79-s + 1.14·83-s − 1.78·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 - 0.267T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 7.46T + 23T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 + 7.46T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 + 0.535T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 1.46T + 97T^{2} \) |
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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
−7.59109428856316679219863144408, −7.00969868785096405011033373952, −6.12841928450477700481523175188, −5.25313403962505451507304695819, −4.43913322235304698008352165747, −3.88537423070610108067820021569, −3.50389188381833146577863243313, −2.02062618112907014712161175196, −1.27069250095171470897952214370, 0,
1.27069250095171470897952214370, 2.02062618112907014712161175196, 3.50389188381833146577863243313, 3.88537423070610108067820021569, 4.43913322235304698008352165747, 5.25313403962505451507304695819, 6.12841928450477700481523175188, 7.00969868785096405011033373952, 7.59109428856316679219863144408
