| L(s) = 1 | − 0.686·2-s + 0.347·3-s − 1.52·4-s − 5-s − 0.238·6-s + 7-s + 2.42·8-s − 2.87·9-s + 0.686·10-s − 0.531·12-s − 6.93·13-s − 0.686·14-s − 0.347·15-s + 1.39·16-s + 1.96·17-s + 1.97·18-s + 6.62·19-s + 1.52·20-s + 0.347·21-s + 0.784·23-s + 0.842·24-s + 25-s + 4.75·26-s − 2.04·27-s − 1.52·28-s + 7.86·29-s + 0.238·30-s + ⋯ |
| L(s) = 1 | − 0.485·2-s + 0.200·3-s − 0.764·4-s − 0.447·5-s − 0.0974·6-s + 0.377·7-s + 0.856·8-s − 0.959·9-s + 0.217·10-s − 0.153·12-s − 1.92·13-s − 0.183·14-s − 0.0897·15-s + 0.348·16-s + 0.475·17-s + 0.465·18-s + 1.51·19-s + 0.341·20-s + 0.0758·21-s + 0.163·23-s + 0.171·24-s + 0.200·25-s + 0.933·26-s − 0.393·27-s − 0.288·28-s + 1.46·29-s + 0.0435·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.686T + 2T^{2} \) |
| 3 | \( 1 - 0.347T + 3T^{2} \) |
| 13 | \( 1 + 6.93T + 13T^{2} \) |
| 17 | \( 1 - 1.96T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 - 0.784T + 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 0.764T + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 + 7.61T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 - 7.75T + 59T^{2} \) |
| 61 | \( 1 + 5.45T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 3.19T + 79T^{2} \) |
| 83 | \( 1 + 3.50T + 83T^{2} \) |
| 89 | \( 1 - 5.50T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.923697951397775921267327416004, −7.68893734420369985840547197003, −6.84128804830153558804365213563, −5.52722616934504985421745201279, −5.06229893932650488888733180147, −4.36402322415156830125981778052, −3.27034936323214654326712050483, −2.54672216547933649385843692012, −1.12352598462094385700107515612, 0,
1.12352598462094385700107515612, 2.54672216547933649385843692012, 3.27034936323214654326712050483, 4.36402322415156830125981778052, 5.06229893932650488888733180147, 5.52722616934504985421745201279, 6.84128804830153558804365213563, 7.68893734420369985840547197003, 7.923697951397775921267327416004
