| L(s) = 1 | + 5-s + 7-s + 5.09·11-s + 2.23·13-s + 5.38·17-s − 19-s − 4.70·23-s − 4·25-s + 5.85·29-s − 4.61·31-s + 35-s + 6.70·37-s + 11.0·41-s − 0.472·43-s + 8.70·47-s + 49-s − 1.32·53-s + 5.09·55-s − 2.70·59-s − 3.47·61-s + 2.23·65-s + 9.09·67-s − 11.1·71-s − 12.0·73-s + 5.09·77-s + 10.9·79-s − 12.3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.53·11-s + 0.620·13-s + 1.30·17-s − 0.229·19-s − 0.981·23-s − 0.800·25-s + 1.08·29-s − 0.829·31-s + 0.169·35-s + 1.10·37-s + 1.73·41-s − 0.0720·43-s + 1.27·47-s + 0.142·49-s − 0.182·53-s + 0.686·55-s − 0.352·59-s − 0.444·61-s + 0.277·65-s + 1.11·67-s − 1.32·71-s − 1.41·73-s + 0.580·77-s + 1.23·79-s − 1.35·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.755382243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.755382243\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 - 5.38T + 17T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 - 8.70T + 47T^{2} \) |
| 53 | \( 1 + 1.32T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 - 9.09T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 4.70T + 97T^{2} \) |
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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
−8.236908792790018649354967695043, −7.64857668808539261312923307109, −6.77011546144461102915573521675, −5.92320012637521954776406096801, −5.68974078030059396919080935511, −4.32098599213049420890103892891, −3.95472121609748790896954782595, −2.86869788755854741994298255701, −1.75137314821791586861668175608, −1.00784907577002794754355377683,
1.00784907577002794754355377683, 1.75137314821791586861668175608, 2.86869788755854741994298255701, 3.95472121609748790896954782595, 4.32098599213049420890103892891, 5.68974078030059396919080935511, 5.92320012637521954776406096801, 6.77011546144461102915573521675, 7.64857668808539261312923307109, 8.236908792790018649354967695043
