| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 11-s − 12-s − 2·13-s + 15-s + 16-s + 17-s + 18-s − 20-s − 22-s + 5.65·23-s − 24-s + 25-s − 2·26-s − 27-s − 7.65·29-s + 30-s − 5.65·31-s + 32-s + 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.223·20-s − 0.213·22-s + 1.17·23-s − 0.204·24-s + 0.200·25-s − 0.392·26-s − 0.192·27-s − 1.42·29-s + 0.182·30-s − 1.01·31-s + 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−7.61999675736610248573624129939, −6.97682742576427042492025123265, −6.34175716904771090287401124688, −5.35368982806586943985244580870, −5.06413793688850314468530698161, −4.13561012452683558319650626772, −3.40686618502901723434424123273, −2.50211085707945536183563892025, −1.37926337551200595808324804532, 0,
1.37926337551200595808324804532, 2.50211085707945536183563892025, 3.40686618502901723434424123273, 4.13561012452683558319650626772, 5.06413793688850314468530698161, 5.35368982806586943985244580870, 6.34175716904771090287401124688, 6.97682742576427042492025123265, 7.61999675736610248573624129939
