L(s) = 1 | − 2-s − 3-s + 4-s − 1.56·5-s + 6-s + 1.56·7-s − 8-s + 9-s + 1.56·10-s + 4.42·11-s − 12-s + 5.02·13-s − 1.56·14-s + 1.56·15-s + 16-s + 17-s − 18-s − 6.32·19-s − 1.56·20-s − 1.56·21-s − 4.42·22-s + 4.03·23-s + 24-s − 2.53·25-s − 5.02·26-s − 27-s + 1.56·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.702·5-s + 0.408·6-s + 0.592·7-s − 0.353·8-s + 0.333·9-s + 0.496·10-s + 1.33·11-s − 0.288·12-s + 1.39·13-s − 0.419·14-s + 0.405·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 1.45·19-s − 0.351·20-s − 0.342·21-s − 0.944·22-s + 0.841·23-s + 0.204·24-s − 0.507·25-s − 0.984·26-s − 0.192·27-s + 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 - 4.03T + 23T^{2} \) |
| 29 | \( 1 + 5.32T + 29T^{2} \) |
| 31 | \( 1 + 9.38T + 31T^{2} \) |
| 37 | \( 1 - 8.41T + 37T^{2} \) |
| 41 | \( 1 + 0.858T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 - 0.373T + 53T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 7.36T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 - 0.0349T + 73T^{2} \) |
| 79 | \( 1 - 3.92T + 79T^{2} \) |
| 83 | \( 1 - 8.34T + 83T^{2} \) |
| 89 | \( 1 - 9.55T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.82632565028592292637238447169, −7.04229633368426702557506366835, −6.36876159401919766265349978296, −5.84408929956564643971976712306, −4.77446942548085288744309798025, −3.94922023753383541383695831750, −3.44090763669101222503380171253, −1.86563163062318753829452358823, −1.26715072178102289804394692514, 0,
1.26715072178102289804394692514, 1.86563163062318753829452358823, 3.44090763669101222503380171253, 3.94922023753383541383695831750, 4.77446942548085288744309798025, 5.84408929956564643971976712306, 6.36876159401919766265349978296, 7.04229633368426702557506366835, 7.82632565028592292637238447169