| L(s) = 1 | + 2.61·2-s + 4.85·4-s + 7.47·8-s + 9.85·16-s + 3.76·17-s − 8.70·19-s − 1.47·23-s + 2.70·31-s + 10.8·32-s + 9.85·34-s − 22.7·38-s − 3.85·46-s − 8.94·47-s − 7·49-s + 14.2·53-s − 14.4·61-s + 7.09·62-s + 8.70·64-s + 18.2·68-s − 42.2·76-s − 14.7·79-s + 11.9·83-s − 7.14·92-s − 23.4·94-s − 18.3·98-s + 37.2·106-s + 17.8·107-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.42·4-s + 2.64·8-s + 2.46·16-s + 0.912·17-s − 1.99·19-s − 0.306·23-s + 0.486·31-s + 1.91·32-s + 1.68·34-s − 3.69·38-s − 0.568·46-s − 1.30·47-s − 49-s + 1.95·53-s − 1.84·61-s + 0.900·62-s + 1.08·64-s + 2.21·68-s − 4.84·76-s − 1.65·79-s + 1.31·83-s − 0.745·92-s − 2.41·94-s − 1.85·98-s + 3.62·106-s + 1.72·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.400390070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.400390070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 + 8.70T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.78038521942540907100259045443, −10.03621504316639370617559622912, −8.558508402508444795291817557582, −7.54860608131123895356374003622, −6.53137444575247663719978475609, −5.92123140089364848706671948637, −4.87442971353380959391737872935, −4.08807451385268501567567573099, −3.08045219779734005587003432611, −1.92860018740434397351518724248,
1.92860018740434397351518724248, 3.08045219779734005587003432611, 4.08807451385268501567567573099, 4.87442971353380959391737872935, 5.92123140089364848706671948637, 6.53137444575247663719978475609, 7.54860608131123895356374003622, 8.558508402508444795291817557582, 10.03621504316639370617559622912, 10.78038521942540907100259045443
