| L(s) = 1 | + 1.75·3-s + 5-s − 7-s + 0.0856·9-s − 3.23·11-s + 13-s + 1.75·15-s − 4.59·17-s + 0.150·19-s − 1.75·21-s + 1.40·23-s + 25-s − 5.11·27-s − 8.67·29-s − 4.96·31-s − 5.68·33-s − 35-s − 0.520·37-s + 1.75·39-s + 3.55·41-s + 0.277·43-s + 0.0856·45-s − 11.1·47-s + 49-s − 8.07·51-s + 0.171·53-s − 3.23·55-s + ⋯ |
| L(s) = 1 | + 1.01·3-s + 0.447·5-s − 0.377·7-s + 0.0285·9-s − 0.975·11-s + 0.277·13-s + 0.453·15-s − 1.11·17-s + 0.0345·19-s − 0.383·21-s + 0.293·23-s + 0.200·25-s − 0.985·27-s − 1.61·29-s − 0.892·31-s − 0.989·33-s − 0.169·35-s − 0.0855·37-s + 0.281·39-s + 0.555·41-s + 0.0422·43-s + 0.0127·45-s − 1.62·47-s + 0.142·49-s − 1.13·51-s + 0.0235·53-s − 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.75T + 3T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 - 0.150T + 19T^{2} \) |
| 23 | \( 1 - 1.40T + 23T^{2} \) |
| 29 | \( 1 + 8.67T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 + 0.520T + 37T^{2} \) |
| 41 | \( 1 - 3.55T + 41T^{2} \) |
| 43 | \( 1 - 0.277T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 0.171T + 53T^{2} \) |
| 59 | \( 1 - 7.78T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 + 1.95T + 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 - 5.85T + 73T^{2} \) |
| 79 | \( 1 + 5.00T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 0.869T + 97T^{2} \) |
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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
−8.301753706830908611609618969551, −7.51541672750358880541433162137, −6.82970035480518123470351836421, −5.85383996586136146947694339092, −5.24586828626391151538739432626, −4.13560308621852390306455175429, −3.30382241927306121630789328208, −2.53277124020744035810263799466, −1.80488282532410733670693829321, 0,
1.80488282532410733670693829321, 2.53277124020744035810263799466, 3.30382241927306121630789328208, 4.13560308621852390306455175429, 5.24586828626391151538739432626, 5.85383996586136146947694339092, 6.82970035480518123470351836421, 7.51541672750358880541433162137, 8.301753706830908611609618969551
