| L(s) = 1 | + 2.30·2-s + 3.30·4-s − 4.30·7-s + 3.00·8-s + 11-s − 5·13-s − 9.90·14-s + 0.302·16-s − 3.90·17-s − 19-s + 2.30·22-s − 3.69·23-s − 11.5·26-s − 14.2·28-s + 9.90·29-s − 4.21·31-s − 5.30·32-s − 9·34-s − 9.60·37-s − 2.30·38-s − 1.60·41-s + 7.21·43-s + 3.30·44-s − 8.51·46-s − 3·47-s + 11.5·49-s − 16.5·52-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.65·4-s − 1.62·7-s + 1.06·8-s + 0.301·11-s − 1.38·13-s − 2.64·14-s + 0.0756·16-s − 0.947·17-s − 0.229·19-s + 0.490·22-s − 0.770·23-s − 2.25·26-s − 2.68·28-s + 1.83·29-s − 0.756·31-s − 0.937·32-s − 1.54·34-s − 1.57·37-s − 0.373·38-s − 0.250·41-s + 1.09·43-s + 0.497·44-s − 1.25·46-s − 0.437·47-s + 1.64·49-s − 2.29·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 7 | \( 1 + 4.30T + 7T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 3.69T + 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 + 9.60T + 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 + 0.211T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 4.60T + 71T^{2} \) |
| 73 | \( 1 + 2.90T + 73T^{2} \) |
| 79 | \( 1 + 0.0916T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 5.30T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.591268366849093939615662148969, −7.25282030245603725073723069555, −6.73660496108074342696780738501, −6.19364040526483233730905915247, −5.31495082266845010545941783590, −4.49289121151963948870230729306, −3.75120802667488387193287067867, −2.91199382599288205957143941015, −2.21239638480208954961775680034, 0,
2.21239638480208954961775680034, 2.91199382599288205957143941015, 3.75120802667488387193287067867, 4.49289121151963948870230729306, 5.31495082266845010545941783590, 6.19364040526483233730905915247, 6.73660496108074342696780738501, 7.25282030245603725073723069555, 8.591268366849093939615662148969
