| L(s) = 1 | − 4.25·2-s + 3·3-s + 10.1·4-s − 22.0·5-s − 12.7·6-s + 8.60·7-s − 8.95·8-s + 9·9-s + 93.9·10-s + 11·11-s + 30.3·12-s − 50.8·13-s − 36.6·14-s − 66.2·15-s − 42.7·16-s − 119.·17-s − 38.2·18-s + 19.0·19-s − 223.·20-s + 25.8·21-s − 46.8·22-s + 173.·23-s − 26.8·24-s + 362.·25-s + 216.·26-s + 27·27-s + 86.9·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 0.577·3-s + 1.26·4-s − 1.97·5-s − 0.868·6-s + 0.464·7-s − 0.395·8-s + 0.333·9-s + 2.97·10-s + 0.301·11-s + 0.729·12-s − 1.08·13-s − 0.699·14-s − 1.14·15-s − 0.667·16-s − 1.70·17-s − 0.501·18-s + 0.230·19-s − 2.49·20-s + 0.268·21-s − 0.453·22-s + 1.57·23-s − 0.228·24-s + 2.90·25-s + 1.63·26-s + 0.192·27-s + 0.587·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 4.25T + 8T^{2} \) |
| 5 | \( 1 + 22.0T + 125T^{2} \) |
| 7 | \( 1 - 8.60T + 343T^{2} \) |
| 13 | \( 1 + 50.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 185.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 8.42T + 2.97e4T^{2} \) |
| 37 | \( 1 - 361.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 99.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 35.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 300.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 363.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 860.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 825.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 26.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 128.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 113.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 209.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 547.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.77e3T + 9.12e5T^{2} \) |
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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
−8.378400260317812878462580291974, −7.79885160506199817319048405897, −7.25488259147657323765268338763, −6.73691123374617412938835093801, −4.77895568539320122282442678352, −4.34627209744276329077645424694, −3.18194753730462708332427661024, −2.17133524963414321260093254784, −0.882365866990857039553933718654, 0,
0.882365866990857039553933718654, 2.17133524963414321260093254784, 3.18194753730462708332427661024, 4.34627209744276329077645424694, 4.77895568539320122282442678352, 6.73691123374617412938835093801, 7.25488259147657323765268338763, 7.79885160506199817319048405897, 8.378400260317812878462580291974
