| L(s) = 1 | − 2-s + 2.25·3-s + 4-s + 1.88·5-s − 2.25·6-s − 3.98·7-s − 8-s + 2.10·9-s − 1.88·10-s − 0.608·11-s + 2.25·12-s + 1.99·13-s + 3.98·14-s + 4.25·15-s + 16-s − 4.77·17-s − 2.10·18-s + 19-s + 1.88·20-s − 9.00·21-s + 0.608·22-s + 6.32·23-s − 2.25·24-s − 1.45·25-s − 1.99·26-s − 2.02·27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.30·3-s + 0.5·4-s + 0.842·5-s − 0.922·6-s − 1.50·7-s − 0.353·8-s + 0.701·9-s − 0.595·10-s − 0.183·11-s + 0.652·12-s + 0.554·13-s + 1.06·14-s + 1.09·15-s + 0.250·16-s − 1.15·17-s − 0.496·18-s + 0.229·19-s + 0.421·20-s − 1.96·21-s + 0.129·22-s + 1.31·23-s − 0.461·24-s − 0.290·25-s − 0.391·26-s − 0.389·27-s − 0.753·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 2.25T + 3T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 + 0.608T + 11T^{2} \) |
| 13 | \( 1 - 1.99T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 23 | \( 1 - 6.32T + 23T^{2} \) |
| 29 | \( 1 + 0.683T + 29T^{2} \) |
| 31 | \( 1 - 3.90T + 31T^{2} \) |
| 37 | \( 1 + 9.81T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 - 3.49T + 43T^{2} \) |
| 47 | \( 1 + 7.01T + 47T^{2} \) |
| 53 | \( 1 + 7.90T + 53T^{2} \) |
| 59 | \( 1 + 0.305T + 59T^{2} \) |
| 61 | \( 1 + 8.75T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 5.49T + 71T^{2} \) |
| 73 | \( 1 - 2.80T + 73T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 + 6.19T + 83T^{2} \) |
| 89 | \( 1 + 2.31T + 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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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
−7.56056157371466415936072142611, −6.87682366791203528340117507776, −6.31417116348942631718024285561, −5.68147891353547761393645206669, −4.52085968006084376309329168172, −3.43338079117121199073809799289, −3.00779987006429872862930599624, −2.31146349006258271021980670118, −1.44424869275162120455896775055, 0,
1.44424869275162120455896775055, 2.31146349006258271021980670118, 3.00779987006429872862930599624, 3.43338079117121199073809799289, 4.52085968006084376309329168172, 5.68147891353547761393645206669, 6.31417116348942631718024285561, 6.87682366791203528340117507776, 7.56056157371466415936072142611
