| L(s) = 1 | + 2-s + 4-s + 0.0881·5-s + 4.92·7-s + 8-s + 0.0881·10-s + 0.970·11-s + 4.90·13-s + 4.92·14-s + 16-s − 3.81·17-s + 2.32·19-s + 0.0881·20-s + 0.970·22-s + 0.0904·23-s − 4.99·25-s + 4.90·26-s + 4.92·28-s − 6.73·29-s + 4.73·31-s + 32-s − 3.81·34-s + 0.434·35-s + 5.77·37-s + 2.32·38-s + 0.0881·40-s − 6.48·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.0394·5-s + 1.86·7-s + 0.353·8-s + 0.0278·10-s + 0.292·11-s + 1.36·13-s + 1.31·14-s + 0.250·16-s − 0.925·17-s + 0.533·19-s + 0.0197·20-s + 0.206·22-s + 0.0188·23-s − 0.998·25-s + 0.962·26-s + 0.931·28-s − 1.24·29-s + 0.850·31-s + 0.176·32-s − 0.654·34-s + 0.0734·35-s + 0.949·37-s + 0.377·38-s + 0.0139·40-s − 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.869940563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.869940563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 0.0881T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 - 0.970T + 11T^{2} \) |
| 13 | \( 1 - 4.90T + 13T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 - 0.0904T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 5.77T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 - 2.46T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 6.91T + 53T^{2} \) |
| 59 | \( 1 - 7.13T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 2.68T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64707399433194061915537101178, −7.27465887019745977021413655895, −6.21461414229816900446335578529, −5.69261629677375180549030840584, −5.02715539921535914147558908299, −4.14408536531510758409168536012, −3.90810282519420402236788811166, −2.60220114708737136586075835398, −1.81011646265159869381670523278, −1.08335600702902202592412261432,
1.08335600702902202592412261432, 1.81011646265159869381670523278, 2.60220114708737136586075835398, 3.90810282519420402236788811166, 4.14408536531510758409168536012, 5.02715539921535914147558908299, 5.69261629677375180549030840584, 6.21461414229816900446335578529, 7.27465887019745977021413655895, 7.64707399433194061915537101178
