| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 7.34·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 14.6·10-s − 51.2·11-s + 12·12-s − 33.6·13-s + 14·14-s + 22.0·15-s + 16·16-s + 107.·17-s − 18·18-s + 57.1·19-s + 29.3·20-s − 21·21-s + 102.·22-s − 23·23-s − 24·24-s − 70.9·25-s + 67.2·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.657·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.464·10-s − 1.40·11-s + 0.288·12-s − 0.717·13-s + 0.267·14-s + 0.379·15-s + 0.250·16-s + 1.53·17-s − 0.235·18-s + 0.690·19-s + 0.328·20-s − 0.218·21-s + 0.993·22-s − 0.208·23-s − 0.204·24-s − 0.567·25-s + 0.507·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 7.34T + 125T^{2} \) |
| 11 | \( 1 + 51.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 57.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 59.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 96.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 389.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 316.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 564.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 266.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 474.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 878.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 189.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 330.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 319.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 931.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 918.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.460492144523251636749505074512, −8.240657624404402649736186617313, −7.79109335429288812021622488880, −6.90338162818058269489740405762, −5.76601793833018279984618479662, −5.02272860474241925319245055921, −3.36357829293459292098709003644, −2.61770339099616940674614498522, −1.51631098170775681564934765798, 0,
1.51631098170775681564934765798, 2.61770339099616940674614498522, 3.36357829293459292098709003644, 5.02272860474241925319245055921, 5.76601793833018279984618479662, 6.90338162818058269489740405762, 7.79109335429288812021622488880, 8.240657624404402649736186617313, 9.460492144523251636749505074512
