| L(s) = 1 | + 2-s − 1.29·3-s + 4-s − 5-s − 1.29·6-s − 4.27·7-s + 8-s − 1.32·9-s − 10-s − 11-s − 1.29·12-s − 0.887·13-s − 4.27·14-s + 1.29·15-s + 16-s − 7.55·17-s − 1.32·18-s − 0.196·19-s − 20-s + 5.52·21-s − 22-s − 4.62·23-s − 1.29·24-s + 25-s − 0.887·26-s + 5.59·27-s − 4.27·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.747·3-s + 0.5·4-s − 0.447·5-s − 0.528·6-s − 1.61·7-s + 0.353·8-s − 0.441·9-s − 0.316·10-s − 0.301·11-s − 0.373·12-s − 0.246·13-s − 1.14·14-s + 0.334·15-s + 0.250·16-s − 1.83·17-s − 0.312·18-s − 0.0450·19-s − 0.223·20-s + 1.20·21-s − 0.213·22-s − 0.963·23-s − 0.264·24-s + 0.200·25-s − 0.174·26-s + 1.07·27-s − 0.806·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1935200634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1935200634\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 1.29T + 3T^{2} \) |
| 7 | \( 1 + 4.27T + 7T^{2} \) |
| 13 | \( 1 + 0.887T + 13T^{2} \) |
| 17 | \( 1 + 7.55T + 17T^{2} \) |
| 19 | \( 1 + 0.196T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 + 8.14T + 29T^{2} \) |
| 31 | \( 1 + 9.85T + 31T^{2} \) |
| 37 | \( 1 + 0.128T + 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 + 1.93T + 43T^{2} \) |
| 47 | \( 1 + 6.59T + 47T^{2} \) |
| 53 | \( 1 - 1.85T + 53T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 + 8.72T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 0.592T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 4.73T + 97T^{2} \) |
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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
−7.51947887392386477510486467479, −6.91569638596795358011958491718, −6.34601521133106090029807521877, −5.78850616647332918260571661406, −5.15798538528710026133089827885, −4.20180090810764930940436595675, −3.65226511977284563903518258693, −2.81945119041250154069564304601, −2.01707286871628116371195220232, −0.18519610125915796983704474741,
0.18519610125915796983704474741, 2.01707286871628116371195220232, 2.81945119041250154069564304601, 3.65226511977284563903518258693, 4.20180090810764930940436595675, 5.15798538528710026133089827885, 5.78850616647332918260571661406, 6.34601521133106090029807521877, 6.91569638596795358011958491718, 7.51947887392386477510486467479
