| L(s) = 1 | − 1.35·3-s + 1.29·5-s − 4.65·7-s − 1.17·9-s + 2.15·11-s − 2.44·13-s − 1.75·15-s − 17-s − 1.10·19-s + 6.29·21-s + 6.39·23-s − 3.31·25-s + 5.64·27-s − 9.24·29-s − 0.493·31-s − 2.91·33-s − 6.05·35-s + 1.57·37-s + 3.31·39-s − 6.56·41-s − 6.61·43-s − 1.52·45-s − 3.72·47-s + 14.7·49-s + 1.35·51-s + 5.94·53-s + 2.80·55-s + ⋯ |
| L(s) = 1 | − 0.780·3-s + 0.581·5-s − 1.76·7-s − 0.391·9-s + 0.650·11-s − 0.679·13-s − 0.453·15-s − 0.242·17-s − 0.254·19-s + 1.37·21-s + 1.33·23-s − 0.662·25-s + 1.08·27-s − 1.71·29-s − 0.0886·31-s − 0.507·33-s − 1.02·35-s + 0.258·37-s + 0.530·39-s − 1.02·41-s − 1.00·43-s − 0.227·45-s − 0.542·47-s + 2.10·49-s + 0.189·51-s + 0.816·53-s + 0.378·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7062071689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7062071689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 5 | \( 1 - 1.29T + 5T^{2} \) |
| 7 | \( 1 + 4.65T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 19 | \( 1 + 1.10T + 19T^{2} \) |
| 23 | \( 1 - 6.39T + 23T^{2} \) |
| 29 | \( 1 + 9.24T + 29T^{2} \) |
| 31 | \( 1 + 0.493T + 31T^{2} \) |
| 37 | \( 1 - 1.57T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 + 6.61T + 43T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 - 5.94T + 53T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 - 0.415T + 71T^{2} \) |
| 73 | \( 1 + 0.107T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 + 1.37T + 83T^{2} \) |
| 89 | \( 1 - 5.85T + 89T^{2} \) |
| 97 | \( 1 - 0.721T + 97T^{2} \) |
| 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
−8.743950564552314322110013955351, −7.43305598347354913492617466925, −6.74552665594027806456962409993, −6.26281575060887434544484888935, −5.62378790706880364670111729105, −4.89835866944770384567094335821, −3.71326005723932789982383432344, −3.03782846059092075951663879095, −1.98735375674818326078531624712, −0.47529978568712610206923928616,
0.47529978568712610206923928616, 1.98735375674818326078531624712, 3.03782846059092075951663879095, 3.71326005723932789982383432344, 4.89835866944770384567094335821, 5.62378790706880364670111729105, 6.26281575060887434544484888935, 6.74552665594027806456962409993, 7.43305598347354913492617466925, 8.743950564552314322110013955351
