| L(s) = 1 | − 2.34·3-s − 5-s + 2.34·7-s + 2.48·9-s + 3.48·11-s − 4.68·13-s + 2.34·15-s − 2.29·17-s − 1.14·19-s − 5.48·21-s − 2.97·23-s + 25-s + 1.19·27-s − 2·29-s + 10.5·31-s − 8.17·33-s − 2.34·35-s − 37-s + 10.9·39-s + 2.17·41-s − 6.97·43-s − 2.48·45-s + 2.34·47-s − 1.51·49-s + 5.37·51-s − 0.217·53-s − 3.48·55-s + ⋯ |
| L(s) = 1 | − 1.35·3-s − 0.447·5-s + 0.885·7-s + 0.829·9-s + 1.05·11-s − 1.29·13-s + 0.604·15-s − 0.556·17-s − 0.262·19-s − 1.19·21-s − 0.621·23-s + 0.200·25-s + 0.230·27-s − 0.371·29-s + 1.88·31-s − 1.42·33-s − 0.396·35-s − 0.164·37-s + 1.75·39-s + 0.339·41-s − 1.06·43-s − 0.371·45-s + 0.341·47-s − 0.215·49-s + 0.752·51-s − 0.0299·53-s − 0.470·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 11 | \( 1 - 3.48T + 11T^{2} \) |
| 13 | \( 1 + 4.68T + 13T^{2} \) |
| 17 | \( 1 + 2.29T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 + 2.97T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 6.97T + 43T^{2} \) |
| 47 | \( 1 - 2.34T + 47T^{2} \) |
| 53 | \( 1 + 0.217T + 53T^{2} \) |
| 59 | \( 1 + 4.22T + 59T^{2} \) |
| 61 | \( 1 - 0.978T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 1.82T + 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 0.393T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.258180792860136767500595293242, −7.55204540811323181984763267248, −6.65308637928003198148137336657, −6.21419112309932402132069421481, −5.06774868920340568260111571558, −4.72074521598873157378686943583, −3.88559166399641248420887387740, −2.44204164324147653900542227125, −1.24660627253951891132263090240, 0,
1.24660627253951891132263090240, 2.44204164324147653900542227125, 3.88559166399641248420887387740, 4.72074521598873157378686943583, 5.06774868920340568260111571558, 6.21419112309932402132069421481, 6.65308637928003198148137336657, 7.55204540811323181984763267248, 8.258180792860136767500595293242
