| L(s) = 1 | + 1.46·2-s − 3-s + 0.158·4-s + 2.14·5-s − 1.46·6-s − 2.70·8-s + 9-s + 3.15·10-s + 2.16·11-s − 0.158·12-s − 3.08·13-s − 2.14·15-s − 4.29·16-s − 0.354·17-s + 1.46·18-s − 4.96·19-s + 0.341·20-s + 3.17·22-s − 23-s + 2.70·24-s − 0.384·25-s − 4.53·26-s − 27-s − 2.50·29-s − 3.15·30-s + 1.83·31-s − 0.896·32-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 0.577·3-s + 0.0794·4-s + 0.960·5-s − 0.599·6-s − 0.956·8-s + 0.333·9-s + 0.998·10-s + 0.652·11-s − 0.0458·12-s − 0.855·13-s − 0.554·15-s − 1.07·16-s − 0.0860·17-s + 0.346·18-s − 1.13·19-s + 0.0763·20-s + 0.677·22-s − 0.208·23-s + 0.552·24-s − 0.0768·25-s − 0.888·26-s − 0.192·27-s − 0.464·29-s − 0.576·30-s + 0.329·31-s − 0.158·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 5 | \( 1 - 2.14T + 5T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 + 3.08T + 13T^{2} \) |
| 17 | \( 1 + 0.354T + 17T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 + 4.50T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.96T + 43T^{2} \) |
| 47 | \( 1 + 7.83T + 47T^{2} \) |
| 53 | \( 1 - 8.86T + 53T^{2} \) |
| 59 | \( 1 + 6.79T + 59T^{2} \) |
| 61 | \( 1 - 2.35T + 61T^{2} \) |
| 67 | \( 1 + 2.20T + 67T^{2} \) |
| 71 | \( 1 - 3.08T + 71T^{2} \) |
| 73 | \( 1 + 3.33T + 73T^{2} \) |
| 79 | \( 1 + 5.15T + 79T^{2} \) |
| 83 | \( 1 + 4.70T + 83T^{2} \) |
| 89 | \( 1 + 1.87T + 89T^{2} \) |
| 97 | \( 1 + 8.12T + 97T^{2} \) |
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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
−8.329211599324096363687555201176, −7.08949125103892439978082649977, −6.47243397147401696659800316878, −5.84106693458235402380967633189, −5.21585263609648716105795427086, −4.48418799615973214194260723012, −3.76079922513237556438913003999, −2.61852093541810308864227753412, −1.69808214608681607290704847497, 0,
1.69808214608681607290704847497, 2.61852093541810308864227753412, 3.76079922513237556438913003999, 4.48418799615973214194260723012, 5.21585263609648716105795427086, 5.84106693458235402380967633189, 6.47243397147401696659800316878, 7.08949125103892439978082649977, 8.329211599324096363687555201176
