| L(s) = 1 | + 13.4·5-s + 33.1·7-s + 11.2·11-s − 57.6·13-s − 22.2·17-s − 19·19-s + 37.1·23-s + 56.7·25-s + 185.·29-s + 304.·31-s + 447.·35-s + 301.·37-s − 138.·41-s + 250.·43-s − 64.8·47-s + 757.·49-s − 707.·53-s + 152.·55-s − 394.·59-s + 294.·61-s − 777.·65-s − 296.·67-s + 528.·71-s + 285.·73-s + 374.·77-s − 581.·79-s + 848.·83-s + ⋯ |
| L(s) = 1 | + 1.20·5-s + 1.79·7-s + 0.309·11-s − 1.23·13-s − 0.317·17-s − 0.229·19-s + 0.336·23-s + 0.453·25-s + 1.18·29-s + 1.76·31-s + 2.15·35-s + 1.34·37-s − 0.527·41-s + 0.888·43-s − 0.201·47-s + 2.20·49-s − 1.83·53-s + 0.372·55-s − 0.869·59-s + 0.618·61-s − 1.48·65-s − 0.540·67-s + 0.882·71-s + 0.457·73-s + 0.553·77-s − 0.828·79-s + 1.12·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.607627086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.607627086\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 - 13.4T + 125T^{2} \) |
| 7 | \( 1 - 33.1T + 343T^{2} \) |
| 11 | \( 1 - 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.2T + 4.91e3T^{2} \) |
| 23 | \( 1 - 37.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 185.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 304.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 250.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 64.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 707.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 394.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 294.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 296.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 528.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 285.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 581.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 848.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.56e3T + 9.12e5T^{2} \) |
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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
−9.299592733525663861569160417808, −8.355447838549689990302273218812, −7.74443307238807480215542079988, −6.69318762768408806827366484862, −5.87356240305736855373106633934, −4.81920202056750060537981918105, −4.55913810038789387748991558626, −2.71602716544162639462550477497, −1.97228426153969677143610506593, −1.01299366883146685475517054966,
1.01299366883146685475517054966, 1.97228426153969677143610506593, 2.71602716544162639462550477497, 4.55913810038789387748991558626, 4.81920202056750060537981918105, 5.87356240305736855373106633934, 6.69318762768408806827366484862, 7.74443307238807480215542079988, 8.355447838549689990302273218812, 9.299592733525663861569160417808
