| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 2.88·5-s − 6·6-s − 22.9·7-s + 8·8-s + 9·9-s + 5.77·10-s + 36.6·11-s − 12·12-s − 36.8·13-s − 45.8·14-s − 8.65·15-s + 16·16-s − 61.7·17-s + 18·18-s − 82.6·19-s + 11.5·20-s + 68.8·21-s + 73.2·22-s + 54.8·23-s − 24·24-s − 116.·25-s − 73.7·26-s − 27·27-s − 91.7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.258·5-s − 0.408·6-s − 1.23·7-s + 0.353·8-s + 0.333·9-s + 0.182·10-s + 1.00·11-s − 0.288·12-s − 0.786·13-s − 0.876·14-s − 0.149·15-s + 0.250·16-s − 0.880·17-s + 0.235·18-s − 0.997·19-s + 0.129·20-s + 0.715·21-s + 0.709·22-s + 0.496·23-s − 0.204·24-s − 0.933·25-s − 0.556·26-s − 0.192·27-s − 0.619·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 59 | \( 1 + 59T \) |
| good | 5 | \( 1 - 2.88T + 125T^{2} \) |
| 7 | \( 1 + 22.9T + 343T^{2} \) |
| 11 | \( 1 - 36.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 82.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 54.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 1.14T + 2.43e4T^{2} \) |
| 31 | \( 1 + 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 70.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 275.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 256.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 23.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 157.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 65.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 378.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 79.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 194.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 225.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 382.T + 9.12e5T^{2} \) |
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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
−10.72710554744033244423174214888, −9.750115067023274959596586314902, −8.949699392025243040662529818399, −7.24258535823399004988838207567, −6.54141329770603555505475555008, −5.78062321782775451242870583830, −4.52306046136334260670538767329, −3.49188783857101113719809190827, −2.00670145175440275668975932967, 0,
2.00670145175440275668975932967, 3.49188783857101113719809190827, 4.52306046136334260670538767329, 5.78062321782775451242870583830, 6.54141329770603555505475555008, 7.24258535823399004988838207567, 8.949699392025243040662529818399, 9.750115067023274959596586314902, 10.72710554744033244423174214888
