| L(s) = 1 | − 5-s + 0.438·7-s − 1.56·11-s − 13-s − 1.56·17-s + 5.12·19-s + 2.43·23-s + 25-s + 7.12·29-s − 6·31-s − 0.438·35-s − 10.6·37-s − 3.56·41-s − 3.12·43-s + 11.1·47-s − 6.80·49-s + 4.68·53-s + 1.56·55-s + 12·59-s − 6.68·61-s + 65-s + 11.3·67-s + 10.4·71-s − 6·73-s − 0.684·77-s − 4.68·79-s − 16.4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.165·7-s − 0.470·11-s − 0.277·13-s − 0.378·17-s + 1.17·19-s + 0.508·23-s + 0.200·25-s + 1.32·29-s − 1.07·31-s − 0.0741·35-s − 1.75·37-s − 0.556·41-s − 0.476·43-s + 1.62·47-s − 0.972·49-s + 0.643·53-s + 0.210·55-s + 1.56·59-s − 0.855·61-s + 0.124·65-s + 1.38·67-s + 1.23·71-s − 0.702·73-s − 0.0780·77-s − 0.527·79-s − 1.81·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.592615242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.592615242\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 0.438T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.68T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 4.68T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−7.54117292379582250177734474009, −7.16892460395433082860188502951, −6.48482262254062153372197414145, −5.36302974769799629566223315223, −5.13894788039934356238748244365, −4.19470506393046977862654132405, −3.40873176592860236043905890097, −2.71379773417269521693085822773, −1.71523426037554875366604296300, −0.60570654296799325185512357845,
0.60570654296799325185512357845, 1.71523426037554875366604296300, 2.71379773417269521693085822773, 3.40873176592860236043905890097, 4.19470506393046977862654132405, 5.13894788039934356238748244365, 5.36302974769799629566223315223, 6.48482262254062153372197414145, 7.16892460395433082860188502951, 7.54117292379582250177734474009
