| L(s) = 1 | − 2-s + 3.15·3-s + 4-s + 5-s − 3.15·6-s + 0.657·7-s − 8-s + 6.97·9-s − 10-s − 4.92·11-s + 3.15·12-s + 3.15·13-s − 0.657·14-s + 3.15·15-s + 16-s − 0.0156·17-s − 6.97·18-s − 1.87·19-s + 20-s + 2.07·21-s + 4.92·22-s + 4.90·23-s − 3.15·24-s + 25-s − 3.15·26-s + 12.5·27-s + 0.657·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.82·3-s + 0.5·4-s + 0.447·5-s − 1.28·6-s + 0.248·7-s − 0.353·8-s + 2.32·9-s − 0.316·10-s − 1.48·11-s + 0.911·12-s + 0.875·13-s − 0.175·14-s + 0.815·15-s + 0.250·16-s − 0.00379·17-s − 1.64·18-s − 0.431·19-s + 0.223·20-s + 0.453·21-s + 1.05·22-s + 1.02·23-s − 0.644·24-s + 0.200·25-s − 0.618·26-s + 2.41·27-s + 0.124·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.428434381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.428434381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 3.15T + 3T^{2} \) |
| 7 | \( 1 - 0.657T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 + 0.0156T + 17T^{2} \) |
| 19 | \( 1 + 1.87T + 19T^{2} \) |
| 23 | \( 1 - 4.90T + 23T^{2} \) |
| 29 | \( 1 - 4.72T + 29T^{2} \) |
| 31 | \( 1 - 6.14T + 31T^{2} \) |
| 37 | \( 1 + 1.63T + 37T^{2} \) |
| 41 | \( 1 + 0.690T + 41T^{2} \) |
| 43 | \( 1 + 0.592T + 43T^{2} \) |
| 47 | \( 1 - 9.37T + 47T^{2} \) |
| 53 | \( 1 - 5.68T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 5.26T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 4.99T + 97T^{2} \) |
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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
−8.283865176236136943447801421105, −7.65620509318750747963543204500, −7.00055984462842498949815022051, −6.17704259311034209029517571029, −5.11735010074774789652629061291, −4.29080947418782011439897641365, −3.20800108160007635772050855919, −2.71708609597354964870567695099, −2.00205806500105077613957702566, −1.03734604050799917321180094847,
1.03734604050799917321180094847, 2.00205806500105077613957702566, 2.71708609597354964870567695099, 3.20800108160007635772050855919, 4.29080947418782011439897641365, 5.11735010074774789652629061291, 6.17704259311034209029517571029, 7.00055984462842498949815022051, 7.65620509318750747963543204500, 8.283865176236136943447801421105
