| L(s) = 1 | − 12.4·5-s + 51.1·11-s + 37.2·13-s − 22.2·17-s + 54.3·19-s − 176.·23-s + 29.9·25-s − 61.0·29-s + 319.·31-s − 315.·37-s + 206.·41-s + 339.·43-s − 142.·47-s − 310.·53-s − 636.·55-s + 281.·59-s − 543.·61-s − 463.·65-s − 479.·67-s − 1.10e3·71-s + 239.·73-s + 1.16e3·79-s + 2.93·83-s + 276.·85-s + 1.27e3·89-s − 676.·95-s + 79.0·97-s + ⋯ |
| L(s) = 1 | − 1.11·5-s + 1.40·11-s + 0.794·13-s − 0.316·17-s + 0.656·19-s − 1.60·23-s + 0.239·25-s − 0.391·29-s + 1.85·31-s − 1.39·37-s + 0.784·41-s + 1.20·43-s − 0.440·47-s − 0.803·53-s − 1.55·55-s + 0.621·59-s − 1.14·61-s − 0.884·65-s − 0.874·67-s − 1.84·71-s + 0.383·73-s + 1.65·79-s + 0.00387·83-s + 0.352·85-s + 1.52·89-s − 0.730·95-s + 0.0827·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.732415038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732415038\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 12.4T + 125T^{2} \) |
| 11 | \( 1 - 51.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 61.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 339.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 142.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 281.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 543.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 479.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 239.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 2.93T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 79.0T + 9.12e5T^{2} \) |
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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
−8.867252081169705071130376864489, −8.121070559850045972099330977573, −7.46994500439216283179085862081, −6.50948730595937040675612361792, −5.90077993846028645910477535601, −4.52761841912903846309456187618, −3.96608009883371703867824881958, −3.20728861047828855995582904470, −1.73600251241790021850459588096, −0.64056766126831727329059723407,
0.64056766126831727329059723407, 1.73600251241790021850459588096, 3.20728861047828855995582904470, 3.96608009883371703867824881958, 4.52761841912903846309456187618, 5.90077993846028645910477535601, 6.50948730595937040675612361792, 7.46994500439216283179085862081, 8.121070559850045972099330977573, 8.867252081169705071130376864489
