| L(s) = 1 | + 5-s − 3.73·7-s + 5.46·11-s + 1.46·13-s − 7.46·17-s + 2·19-s + 0.267·23-s + 25-s − 8.46·29-s + 2·31-s − 3.73·35-s − 10.3·37-s + 3.92·41-s + 11.4·43-s + 3.73·47-s + 6.92·49-s − 6·53-s + 5.46·55-s − 6.39·59-s − 1.53·61-s + 1.46·65-s − 9.73·67-s + 2.53·71-s − 6.92·73-s − 20.3·77-s − 8.53·79-s − 2.80·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.41·7-s + 1.64·11-s + 0.406·13-s − 1.81·17-s + 0.458·19-s + 0.0558·23-s + 0.200·25-s − 1.57·29-s + 0.359·31-s − 0.630·35-s − 1.70·37-s + 0.613·41-s + 1.74·43-s + 0.544·47-s + 0.989·49-s − 0.824·53-s + 0.736·55-s − 0.832·59-s − 0.196·61-s + 0.181·65-s − 1.18·67-s + 0.300·71-s − 0.810·73-s − 2.32·77-s − 0.960·79-s − 0.307·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 0.267T + 23T^{2} \) |
| 29 | \( 1 + 8.46T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 6.39T + 59T^{2} \) |
| 61 | \( 1 + 1.53T + 61T^{2} \) |
| 67 | \( 1 + 9.73T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 + 2.80T + 83T^{2} \) |
| 89 | \( 1 + 3.92T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
| 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
−7.35540061893367213706621863239, −6.93243661603426553001860792585, −6.11600546646132684633593309793, −5.95468938784116343163096009056, −4.65107677838850356625072135231, −3.91554265887523293236633708871, −3.29761567811302235412314330432, −2.29465623888969538217371935551, −1.33616080835037379873265127108, 0,
1.33616080835037379873265127108, 2.29465623888969538217371935551, 3.29761567811302235412314330432, 3.91554265887523293236633708871, 4.65107677838850356625072135231, 5.95468938784116343163096009056, 6.11600546646132684633593309793, 6.93243661603426553001860792585, 7.35540061893367213706621863239
