| L(s) = 1 | + 2-s + 3-s + 4-s − 0.554·5-s + 6-s − 7-s + 8-s + 9-s − 0.554·10-s − 1.64·11-s + 12-s − 14-s − 0.554·15-s + 16-s + 2.91·17-s + 18-s − 6.85·19-s − 0.554·20-s − 21-s − 1.64·22-s + 2.93·23-s + 24-s − 4.69·25-s + 27-s − 28-s − 3.69·29-s − 0.554·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.248·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.175·10-s − 0.495·11-s + 0.288·12-s − 0.267·14-s − 0.143·15-s + 0.250·16-s + 0.706·17-s + 0.235·18-s − 1.57·19-s − 0.124·20-s − 0.218·21-s − 0.350·22-s + 0.612·23-s + 0.204·24-s − 0.938·25-s + 0.192·27-s − 0.188·28-s − 0.685·29-s − 0.101·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.554T + 5T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 + 6.85T + 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 + 7.93T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 5.13T + 53T^{2} \) |
| 59 | \( 1 - 0.158T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 1.69T + 67T^{2} \) |
| 71 | \( 1 + 1.01T + 71T^{2} \) |
| 73 | \( 1 + 5.97T + 73T^{2} \) |
| 79 | \( 1 - 8.92T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.58275326297485188127091633018, −6.80418033090173681594354727979, −6.25910323116966931840198714223, −5.32671676692282376846304009956, −4.73532218459427943069136563771, −3.74903534729400470700548729982, −3.37381194477403119989888635330, −2.40494773982222444427345551214, −1.63120940171904312154673768288, 0,
1.63120940171904312154673768288, 2.40494773982222444427345551214, 3.37381194477403119989888635330, 3.74903534729400470700548729982, 4.73532218459427943069136563771, 5.32671676692282376846304009956, 6.25910323116966931840198714223, 6.80418033090173681594354727979, 7.58275326297485188127091633018
