| L(s) = 1 | − 1.08·3-s − 5-s + 7-s − 1.81·9-s − 4.28·11-s − 1.53·13-s + 1.08·15-s + 4.46·17-s − 1.61·19-s − 1.08·21-s − 23-s + 25-s + 5.24·27-s + 8.54·29-s + 1.12·31-s + 4.66·33-s − 35-s + 2.71·37-s + 1.67·39-s − 8.87·41-s + 10.7·43-s + 1.81·45-s + 8.25·47-s + 49-s − 4.86·51-s + 7.50·53-s + 4.28·55-s + ⋯ |
| L(s) = 1 | − 0.628·3-s − 0.447·5-s + 0.377·7-s − 0.604·9-s − 1.29·11-s − 0.426·13-s + 0.281·15-s + 1.08·17-s − 0.370·19-s − 0.237·21-s − 0.208·23-s + 0.200·25-s + 1.00·27-s + 1.58·29-s + 0.201·31-s + 0.812·33-s − 0.169·35-s + 0.446·37-s + 0.268·39-s − 1.38·41-s + 1.64·43-s + 0.270·45-s + 1.20·47-s + 0.142·49-s − 0.680·51-s + 1.03·53-s + 0.577·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.08T + 3T^{2} \) |
| 11 | \( 1 + 4.28T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 + 1.61T + 19T^{2} \) |
| 29 | \( 1 - 8.54T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 + 8.87T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 8.25T + 47T^{2} \) |
| 53 | \( 1 - 7.50T + 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 7.25T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 2.61T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−7.64267436646890190158816050615, −7.09369106315836757033873921284, −6.04578794336923017163655596441, −5.56043180305964720034548522441, −4.87716438287313152696654155738, −4.20665457392917476825571171245, −3.01338353674184365691958015120, −2.50586528110271900708993854683, −1.05658994283121408683647180543, 0,
1.05658994283121408683647180543, 2.50586528110271900708993854683, 3.01338353674184365691958015120, 4.20665457392917476825571171245, 4.87716438287313152696654155738, 5.56043180305964720034548522441, 6.04578794336923017163655596441, 7.09369106315836757033873921284, 7.64267436646890190158816050615
