| L(s) = 1 | − 2i·2-s − 4·4-s − 15.1i·7-s + 8i·8-s + 34.3·11-s + 18.1i·13-s − 30.3·14-s + 16·16-s − 46.3i·17-s − 43.1·19-s − 68.7i·22-s − 20.7i·23-s + 36.3·26-s + 60.7i·28-s + 58.3·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.819i·7-s + 0.353i·8-s + 0.942·11-s + 0.387i·13-s − 0.579·14-s + 0.250·16-s − 0.661i·17-s − 0.521·19-s − 0.666i·22-s − 0.188i·23-s + 0.274·26-s + 0.409i·28-s + 0.373·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.597958521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597958521\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 15.1iT - 343T^{2} \) |
| 11 | \( 1 - 34.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 46.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 43.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 20.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 58.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 30.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 145. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 115.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 317. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 375. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 119. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 161.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 892.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 289. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 702.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 434. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 259T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3iT - 9.12e5T^{2} \) |
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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
−9.002128139863339548106097639822, −8.291610211952985304261096145717, −7.16283687998871883132527836707, −6.57654338228858704560598937700, −5.36303249304395732935007747803, −4.32379009106807816318677831636, −3.77400953950211517213130134867, −2.58113428504070179206041344890, −1.42672484593955734795086531160, −0.41362054448941331859228846517,
1.14214959887977250782530211598, 2.47444034339802652334367476259, 3.70864558615097883075193906288, 4.58357659425470976487060491594, 5.66403152846029951089865148945, 6.20881737627533355873758788103, 7.05829831315555120926962412536, 8.050772574648647877040109021622, 8.690430850217269578661116297024, 9.360690232842894040289419632061
