| L(s) = 1 | + 2.38·2-s + 3.68·4-s − 0.998·5-s − 3.46·7-s + 4.01·8-s − 2.37·10-s + 1.23·11-s − 0.762·13-s − 8.26·14-s + 2.20·16-s + 2.76·17-s − 6.45·19-s − 3.67·20-s + 2.94·22-s − 4.57·23-s − 4.00·25-s − 1.81·26-s − 12.7·28-s + 6.92·29-s + 1.57·31-s − 2.77·32-s + 6.58·34-s + 3.46·35-s + 1.00·37-s − 15.3·38-s − 4.00·40-s − 2.61·41-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.84·4-s − 0.446·5-s − 1.31·7-s + 1.41·8-s − 0.752·10-s + 0.373·11-s − 0.211·13-s − 2.20·14-s + 0.550·16-s + 0.670·17-s − 1.48·19-s − 0.822·20-s + 0.628·22-s − 0.953·23-s − 0.800·25-s − 0.356·26-s − 2.41·28-s + 1.28·29-s + 0.283·31-s − 0.491·32-s + 1.12·34-s + 0.584·35-s + 0.164·37-s − 2.49·38-s − 0.633·40-s − 0.408·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 5 | \( 1 + 0.998T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 0.762T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 1.57T + 31T^{2} \) |
| 37 | \( 1 - 1.00T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 3.40T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + 0.0539T + 67T^{2} \) |
| 71 | \( 1 + 5.95T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 7.61T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.899887707363032212959552893642, −6.90727233177616640450601823844, −6.36788578528701120848193595259, −5.96422759209355629209448185814, −4.89609628485861282485190087238, −4.21702360686845698474242409256, −3.52356744301379595255336480082, −2.93970634228474698676762683911, −1.89462489914987081837972456238, 0,
1.89462489914987081837972456238, 2.93970634228474698676762683911, 3.52356744301379595255336480082, 4.21702360686845698474242409256, 4.89609628485861282485190087238, 5.96422759209355629209448185814, 6.36788578528701120848193595259, 6.90727233177616640450601823844, 7.899887707363032212959552893642
