| L(s) = 1 | + 2.58·2-s − 0.518·3-s + 4.70·4-s − 1.61·5-s − 1.34·6-s + 6.99·8-s − 2.73·9-s − 4.17·10-s + 2.70·11-s − 2.43·12-s + 0.835·15-s + 8.69·16-s − 3.12·17-s − 7.06·18-s − 3.68·19-s − 7.57·20-s + 7.00·22-s + 1.98·23-s − 3.62·24-s − 2.40·25-s + 2.97·27-s − 5.37·29-s + 2.16·30-s − 10.4·31-s + 8.52·32-s − 1.40·33-s − 8.09·34-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 0.299·3-s + 2.35·4-s − 0.720·5-s − 0.547·6-s + 2.47·8-s − 0.910·9-s − 1.31·10-s + 0.815·11-s − 0.703·12-s + 0.215·15-s + 2.17·16-s − 0.758·17-s − 1.66·18-s − 0.844·19-s − 1.69·20-s + 1.49·22-s + 0.414·23-s − 0.739·24-s − 0.480·25-s + 0.571·27-s − 0.997·29-s + 0.395·30-s − 1.88·31-s + 1.50·32-s − 0.244·33-s − 1.38·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 + 0.518T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 + 1.05T + 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 + 9.10T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 + 1.75T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.22185745787224736399027089329, −6.49121023750301677983817740827, −5.90748414262251256813086208822, −5.41387963355375387246745050630, −4.43068346481761444574908797922, −4.08541330388571153416781247309, −3.35846809142187809212350088176, −2.55780157832447978600363372478, −1.67138669880236841740976261235, 0,
1.67138669880236841740976261235, 2.55780157832447978600363372478, 3.35846809142187809212350088176, 4.08541330388571153416781247309, 4.43068346481761444574908797922, 5.41387963355375387246745050630, 5.90748414262251256813086208822, 6.49121023750301677983817740827, 7.22185745787224736399027089329
