| L(s) = 1 | − 2-s + 2.62·3-s + 4-s + 0.617·5-s − 2.62·6-s − 4.72·7-s − 8-s + 3.91·9-s − 0.617·10-s − 0.448·11-s + 2.62·12-s − 5.04·13-s + 4.72·14-s + 1.62·15-s + 16-s + 5.77·17-s − 3.91·18-s + 7.66·19-s + 0.617·20-s − 12.4·21-s + 0.448·22-s − 2.88·23-s − 2.62·24-s − 4.61·25-s + 5.04·26-s + 2.40·27-s − 4.72·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.51·3-s + 0.5·4-s + 0.276·5-s − 1.07·6-s − 1.78·7-s − 0.353·8-s + 1.30·9-s − 0.195·10-s − 0.135·11-s + 0.759·12-s − 1.39·13-s + 1.26·14-s + 0.419·15-s + 0.250·16-s + 1.40·17-s − 0.922·18-s + 1.75·19-s + 0.138·20-s − 2.71·21-s + 0.0956·22-s − 0.601·23-s − 0.536·24-s − 0.923·25-s + 0.989·26-s + 0.463·27-s − 0.893·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 - 0.617T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 11 | \( 1 + 0.448T + 11T^{2} \) |
| 13 | \( 1 + 5.04T + 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 + 2.88T + 23T^{2} \) |
| 29 | \( 1 + 0.550T + 29T^{2} \) |
| 31 | \( 1 + 0.109T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 + 2.63T + 47T^{2} \) |
| 53 | \( 1 + 8.41T + 53T^{2} \) |
| 59 | \( 1 + 0.494T + 59T^{2} \) |
| 61 | \( 1 - 6.95T + 61T^{2} \) |
| 67 | \( 1 - 2.04T + 67T^{2} \) |
| 71 | \( 1 + 9.08T + 71T^{2} \) |
| 73 | \( 1 + 3.05T + 73T^{2} \) |
| 79 | \( 1 - 0.0912T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 + 5.71T + 97T^{2} \) |
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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
−8.095215111609806645118124231679, −7.42124974088297056503590591600, −7.05565551377779840051704576144, −5.99649773342039283521208774767, −5.21142005038632604994841225135, −3.70346073789115647691851322257, −3.18593922057663592137718762953, −2.64259189259908806233796686431, −1.58296171193663368915858858930, 0,
1.58296171193663368915858858930, 2.64259189259908806233796686431, 3.18593922057663592137718762953, 3.70346073789115647691851322257, 5.21142005038632604994841225135, 5.99649773342039283521208774767, 7.05565551377779840051704576144, 7.42124974088297056503590591600, 8.095215111609806645118124231679
