| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2.52·7-s + 8-s + 9-s − 10-s − 2.04·11-s − 12-s − 0.223·13-s + 2.52·14-s + 15-s + 16-s + 18-s + 1.73·19-s − 20-s − 2.52·21-s − 2.04·22-s − 7.65·23-s − 24-s + 25-s − 0.223·26-s − 27-s + 2.52·28-s − 7.09·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.953·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.616·11-s − 0.288·12-s − 0.0619·13-s + 0.674·14-s + 0.258·15-s + 0.250·16-s + 0.235·18-s + 0.398·19-s − 0.223·20-s − 0.550·21-s − 0.435·22-s − 1.59·23-s − 0.204·24-s + 0.200·25-s − 0.0438·26-s − 0.192·27-s + 0.476·28-s − 1.31·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 + 0.223T + 13T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 + 7.09T + 29T^{2} \) |
| 31 | \( 1 - 5.71T + 31T^{2} \) |
| 37 | \( 1 - 2.65T + 37T^{2} \) |
| 41 | \( 1 + 5.55T + 41T^{2} \) |
| 43 | \( 1 + 0.420T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 + 5.57T + 53T^{2} \) |
| 59 | \( 1 + 9.10T + 59T^{2} \) |
| 61 | \( 1 - 3.62T + 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 3.03T + 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 - 1.47T + 83T^{2} \) |
| 89 | \( 1 + 6.98T + 89T^{2} \) |
| 97 | \( 1 + 7.35T + 97T^{2} \) |
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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
−7.61498419677034008040237464833, −6.60284997019449227361522497492, −5.87986796372152362363152358597, −5.32690979069664364035082076353, −4.60752560072827479093021042896, −4.11316330239353225487606031017, −3.19832773351211143772360046284, −2.21231692728229099164716727076, −1.37425376565843649057157984670, 0,
1.37425376565843649057157984670, 2.21231692728229099164716727076, 3.19832773351211143772360046284, 4.11316330239353225487606031017, 4.60752560072827479093021042896, 5.32690979069664364035082076353, 5.87986796372152362363152358597, 6.60284997019449227361522497492, 7.61498419677034008040237464833
