| L(s) = 1 | − 4.26·2-s + 3·3-s + 10.1·4-s − 7.59·5-s − 12.7·6-s − 6.39·7-s − 9.18·8-s + 9·9-s + 32.3·10-s − 11·11-s + 30.4·12-s + 49.6·13-s + 27.2·14-s − 22.7·15-s − 42.1·16-s + 72.0·17-s − 38.3·18-s + 142.·19-s − 77.1·20-s − 19.1·21-s + 46.8·22-s + 162.·23-s − 27.5·24-s − 67.3·25-s − 211.·26-s + 27·27-s − 64.9·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 0.577·3-s + 1.26·4-s − 0.679·5-s − 0.869·6-s − 0.345·7-s − 0.405·8-s + 0.333·9-s + 1.02·10-s − 0.301·11-s + 0.732·12-s + 1.05·13-s + 0.519·14-s − 0.392·15-s − 0.657·16-s + 1.02·17-s − 0.502·18-s + 1.71·19-s − 0.862·20-s − 0.199·21-s + 0.454·22-s + 1.47·23-s − 0.234·24-s − 0.538·25-s − 1.59·26-s + 0.192·27-s − 0.438·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.280871386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280871386\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 4.26T + 8T^{2} \) |
| 5 | \( 1 + 7.59T + 125T^{2} \) |
| 7 | \( 1 + 6.39T + 343T^{2} \) |
| 13 | \( 1 - 49.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 162.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 209.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 10.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 28.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 13.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 283.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 17.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 198.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 205.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 332.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 191.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 576.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 574.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 636.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 567.T + 9.12e5T^{2} \) |
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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
−8.753907624334756960815447018679, −8.117778930885930868555281363927, −7.53470335900372790748290123235, −6.94653106122408128647449200576, −5.83108899086517874015932188269, −4.66570260122332408827083244574, −3.47146037831953579449604272286, −2.84257844654185634052854734683, −1.36622366137483765772488192630, −0.72564206252996149669862509069,
0.72564206252996149669862509069, 1.36622366137483765772488192630, 2.84257844654185634052854734683, 3.47146037831953579449604272286, 4.66570260122332408827083244574, 5.83108899086517874015932188269, 6.94653106122408128647449200576, 7.53470335900372790748290123235, 8.117778930885930868555281363927, 8.753907624334756960815447018679
