| L(s) = 1 | − 2.62·3-s − 5-s − 2.55·7-s + 3.91·9-s − 2.46·11-s + 1.55·13-s + 2.62·15-s − 6.83·17-s + 7.66·19-s + 6.70·21-s + 7.50·23-s + 25-s − 2.39·27-s + 3.25·29-s − 0.658·31-s + 6.48·33-s + 2.55·35-s + 37-s − 4.09·39-s + 2.46·41-s − 10.9·43-s − 3.91·45-s − 3.11·47-s − 0.485·49-s + 17.9·51-s + 8.64·53-s + 2.46·55-s + ⋯ |
| L(s) = 1 | − 1.51·3-s − 0.447·5-s − 0.964·7-s + 1.30·9-s − 0.744·11-s + 0.432·13-s + 0.678·15-s − 1.65·17-s + 1.75·19-s + 1.46·21-s + 1.56·23-s + 0.200·25-s − 0.460·27-s + 0.604·29-s − 0.118·31-s + 1.12·33-s + 0.431·35-s + 0.164·37-s − 0.656·39-s + 0.385·41-s − 1.67·43-s − 0.582·45-s − 0.454·47-s − 0.0692·49-s + 2.51·51-s + 1.18·53-s + 0.332·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.62T + 3T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + 2.46T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 + 6.83T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + 0.658T + 31T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 3.11T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 - 6.23T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 - 8.06T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−8.396959140388180474020591550573, −7.24188732254505740092564111774, −6.81640709496995942122297468158, −6.12214569637732916425932733372, −5.22061096574823682298447574406, −4.77389846871387830475818770869, −3.61709300233872081136384485912, −2.71949764687588940468497337979, −1.02909451604647836276138012946, 0,
1.02909451604647836276138012946, 2.71949764687588940468497337979, 3.61709300233872081136384485912, 4.77389846871387830475818770869, 5.22061096574823682298447574406, 6.12214569637732916425932733372, 6.81640709496995942122297468158, 7.24188732254505740092564111774, 8.396959140388180474020591550573
