| L(s) = 1 | − 2-s − 0.538·3-s + 4-s + 5-s + 0.538·6-s − 8-s − 2.70·9-s − 10-s + 0.568·11-s − 0.538·12-s − 0.762·13-s − 0.538·15-s + 16-s + 17-s + 2.70·18-s + 1.64·19-s + 20-s − 0.568·22-s + 4.70·23-s + 0.538·24-s + 25-s + 0.762·26-s + 3.07·27-s + 1.83·29-s + 0.538·30-s − 5.81·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.311·3-s + 0.5·4-s + 0.447·5-s + 0.220·6-s − 0.353·8-s − 0.903·9-s − 0.316·10-s + 0.171·11-s − 0.155·12-s − 0.211·13-s − 0.139·15-s + 0.250·16-s + 0.242·17-s + 0.638·18-s + 0.378·19-s + 0.223·20-s − 0.121·22-s + 0.981·23-s + 0.110·24-s + 0.200·25-s + 0.149·26-s + 0.592·27-s + 0.340·29-s + 0.0983·30-s − 1.04·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 0.538T + 3T^{2} \) |
| 11 | \( 1 - 0.568T + 11T^{2} \) |
| 13 | \( 1 + 0.762T + 13T^{2} \) |
| 19 | \( 1 - 1.64T + 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 + 5.81T + 31T^{2} \) |
| 37 | \( 1 + 9.68T + 37T^{2} \) |
| 41 | \( 1 - 2.72T + 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 5.88T + 53T^{2} \) |
| 59 | \( 1 + 7.96T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 3.35T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 + 0.559T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 8.39T + 89T^{2} \) |
| 97 | \( 1 + 3.59T + 97T^{2} \) |
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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
−7.46877807297855453684963160485, −6.75355084842579246189785458323, −6.23979897993626879043925714730, −5.30361567011987815980041956128, −5.04144320849584778516193228485, −3.64712185217725717214951572702, −2.98611573378902291408123564700, −2.08880567960249192541715959406, −1.14274936800527076918652435876, 0,
1.14274936800527076918652435876, 2.08880567960249192541715959406, 2.98611573378902291408123564700, 3.64712185217725717214951572702, 5.04144320849584778516193228485, 5.30361567011987815980041956128, 6.23979897993626879043925714730, 6.75355084842579246189785458323, 7.46877807297855453684963160485
