| L(s) = 1 | − 5-s − 2.91·7-s + 11-s + 3.43·13-s − 4.91·17-s + 4.35·19-s + 4.35·23-s + 25-s − 2·29-s + 6.35·31-s + 2.91·35-s + 2·37-s − 2·41-s − 7.27·43-s − 0.354·47-s + 1.51·49-s − 7.83·53-s − 55-s + 6.35·59-s − 15.0·61-s − 3.43·65-s − 1.83·67-s − 0.872·71-s − 10.7·73-s − 2.91·77-s − 2.87·79-s − 17.4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.10·7-s + 0.301·11-s + 0.953·13-s − 1.19·17-s + 0.999·19-s + 0.908·23-s + 0.200·25-s − 0.371·29-s + 1.14·31-s + 0.493·35-s + 0.328·37-s − 0.312·41-s − 1.10·43-s − 0.0517·47-s + 0.216·49-s − 1.07·53-s − 0.134·55-s + 0.827·59-s − 1.92·61-s − 0.426·65-s − 0.224·67-s − 0.103·71-s − 1.25·73-s − 0.332·77-s − 0.323·79-s − 1.91·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.417619071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417619071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2.91T + 7T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 17 | \( 1 + 4.91T + 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 23 | \( 1 - 4.35T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 7.27T + 43T^{2} \) |
| 47 | \( 1 + 0.354T + 47T^{2} \) |
| 53 | \( 1 + 7.83T + 53T^{2} \) |
| 59 | \( 1 - 6.35T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 + 1.83T + 67T^{2} \) |
| 71 | \( 1 + 0.872T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 2.87T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.76784692966193240770269338095, −7.08829910622908598429026700969, −6.42923900041193145096099218526, −6.00607031873828036620076040268, −4.92186902152704414789994707432, −4.28713143687582927553303061746, −3.31197152145623431429711060555, −3.02496804240663165493719755301, −1.69169927610135160158495235336, −0.59546545722128343648614875853,
0.59546545722128343648614875853, 1.69169927610135160158495235336, 3.02496804240663165493719755301, 3.31197152145623431429711060555, 4.28713143687582927553303061746, 4.92186902152704414789994707432, 6.00607031873828036620076040268, 6.42923900041193145096099218526, 7.08829910622908598429026700969, 7.76784692966193240770269338095
