| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 1.73·7-s + 8-s + 9-s + 10-s − 12-s + 0.464·13-s − 1.73·14-s − 15-s + 16-s − 7.73·17-s + 18-s + 0.464·19-s + 20-s + 1.73·21-s − 2.19·23-s − 24-s + 25-s + 0.464·26-s − 27-s − 1.73·28-s + 0.464·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.654·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s + 0.128·13-s − 0.462·14-s − 0.258·15-s + 0.250·16-s − 1.87·17-s + 0.235·18-s + 0.106·19-s + 0.223·20-s + 0.377·21-s − 0.457·23-s − 0.204·24-s + 0.200·25-s + 0.0910·26-s − 0.192·27-s − 0.327·28-s + 0.0861·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 - 0.464T + 13T^{2} \) |
| 17 | \( 1 + 7.73T + 17T^{2} \) |
| 19 | \( 1 - 0.464T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 0.464T + 29T^{2} \) |
| 31 | \( 1 - 0.196T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 1.26T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 8.19T + 53T^{2} \) |
| 59 | \( 1 + 8.19T + 59T^{2} \) |
| 61 | \( 1 + 4.73T + 61T^{2} \) |
| 67 | \( 1 + 6.39T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 4.39T + 89T^{2} \) |
| 97 | \( 1 + 0.196T + 97T^{2} \) |
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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
−8.126546771804876319163243814624, −7.03190620369022893967166947646, −6.53089740743057091084277917994, −6.02119886732138844052808609551, −5.10448638917216533959884876912, −4.46018989788031575541258359894, −3.56336897420761927912411027673, −2.57353098095275865165500781365, −1.62637966065797898038249460880, 0,
1.62637966065797898038249460880, 2.57353098095275865165500781365, 3.56336897420761927912411027673, 4.46018989788031575541258359894, 5.10448638917216533959884876912, 6.02119886732138844052808609551, 6.53089740743057091084277917994, 7.03190620369022893967166947646, 8.126546771804876319163243814624
