| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 17.3·5-s − 6·6-s + 29.1·7-s + 8·8-s + 9·9-s + 34.7·10-s + 34.2·11-s − 12·12-s − 37.9·13-s + 58.2·14-s − 52.1·15-s + 16·16-s − 84.9·17-s + 18·18-s − 8.58·19-s + 69.4·20-s − 87.4·21-s + 68.5·22-s − 103.·23-s − 24·24-s + 176.·25-s − 75.8·26-s − 27·27-s + 116.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.55·5-s − 0.408·6-s + 1.57·7-s + 0.353·8-s + 0.333·9-s + 1.09·10-s + 0.939·11-s − 0.288·12-s − 0.808·13-s + 1.11·14-s − 0.897·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s − 0.103·19-s + 0.776·20-s − 0.908·21-s + 0.663·22-s − 0.934·23-s − 0.204·24-s + 1.41·25-s − 0.571·26-s − 0.192·27-s + 0.786·28-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
| \(L(2)\) |
\(\approx\) |
\(3.770942776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.770942776\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 17.3T + 125T^{2} \) |
| 7 | \( 1 - 29.1T + 343T^{2} \) |
| 11 | \( 1 - 34.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 8.58T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 173.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 346.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 26.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 472.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 514.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 133.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 222.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 950.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 183.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 654.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 422.T + 9.12e5T^{2} \) |
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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
−11.17831389750332609387167767334, −10.30738664724413695253197374733, −9.366890753362849724807963516134, −8.174666106655518455851014337995, −6.80770070532304743466576053444, −6.08409200164414383647167449692, −5.04551120434003619070125149114, −4.41528676509960890890291669222, −2.31767093845553973520191968782, −1.46972330159075926557417111432,
1.46972330159075926557417111432, 2.31767093845553973520191968782, 4.41528676509960890890291669222, 5.04551120434003619070125149114, 6.08409200164414383647167449692, 6.80770070532304743466576053444, 8.174666106655518455851014337995, 9.366890753362849724807963516134, 10.30738664724413695253197374733, 11.17831389750332609387167767334
