| L(s) = 1 | + 1.56·3-s − 3.56·5-s − 0.561·9-s − 11-s − 2·13-s − 5.56·15-s − 1.12·17-s − 7.12·19-s + 4.68·23-s + 7.68·25-s − 5.56·27-s + 1.12·29-s − 9.56·31-s − 1.56·33-s − 6.68·37-s − 3.12·39-s + 8.24·41-s − 7.12·43-s + 2·45-s − 4·47-s − 7·49-s − 1.75·51-s + 8.24·53-s + 3.56·55-s − 11.1·57-s + 12.6·59-s + 15.3·61-s + ⋯ |
| L(s) = 1 | + 0.901·3-s − 1.59·5-s − 0.187·9-s − 0.301·11-s − 0.554·13-s − 1.43·15-s − 0.272·17-s − 1.63·19-s + 0.976·23-s + 1.53·25-s − 1.07·27-s + 0.208·29-s − 1.71·31-s − 0.271·33-s − 1.09·37-s − 0.500·39-s + 1.28·41-s − 1.08·43-s + 0.298·45-s − 0.583·47-s − 49-s − 0.245·51-s + 1.13·53-s + 0.480·55-s − 1.47·57-s + 1.65·59-s + 1.96·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 9.56T + 31T^{2} \) |
| 37 | \( 1 + 6.68T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 - 3.31T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 + 6.68T + 97T^{2} \) |
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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
−9.965088507638647073968732990367, −8.656929244624234630824060750928, −8.530945589208049100672885995963, −7.49674853250841138701759634638, −6.84666074039592107350230004766, −5.28888958868363812420471932101, −4.17676354055744813843210672677, −3.41502641256963178080103047658, −2.30906917918481945396602919838, 0,
2.30906917918481945396602919838, 3.41502641256963178080103047658, 4.17676354055744813843210672677, 5.28888958868363812420471932101, 6.84666074039592107350230004766, 7.49674853250841138701759634638, 8.530945589208049100672885995963, 8.656929244624234630824060750928, 9.965088507638647073968732990367
