| L(s) = 1 | − 25·5-s − 105.·7-s + 447.·11-s + 276.·13-s − 1.82e3·17-s + 1.37e3·19-s − 1.12e3·23-s + 625·25-s − 1.62e3·29-s + 443.·31-s + 2.64e3·35-s + 1.25e4·37-s − 1.68e3·41-s + 8.86e3·43-s + 2.77e3·47-s − 5.61e3·49-s + 3.01e4·53-s − 1.11e4·55-s − 3.31e4·59-s + 2.59e4·61-s − 6.90e3·65-s + 1.93e4·67-s − 5.28e4·71-s + 3.57e4·73-s − 4.73e4·77-s − 9.18e4·79-s − 2.02e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.816·7-s + 1.11·11-s + 0.453·13-s − 1.53·17-s + 0.871·19-s − 0.442·23-s + 0.200·25-s − 0.357·29-s + 0.0829·31-s + 0.364·35-s + 1.51·37-s − 0.156·41-s + 0.731·43-s + 0.183·47-s − 0.334·49-s + 1.47·53-s − 0.498·55-s − 1.23·59-s + 0.893·61-s − 0.202·65-s + 0.527·67-s − 1.24·71-s + 0.784·73-s − 0.910·77-s − 1.65·79-s − 0.323·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 7 | \( 1 + 105.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 447.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 276.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.82e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.37e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.62e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 443.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.25e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.68e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.86e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.77e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.01e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.02e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
| 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
−9.220135221196649089483314312911, −8.498305487193909611392596711711, −7.34994301157373200405754816492, −6.59206106647312839868292912255, −5.82352281033431510048667037285, −4.40501873903252195270502828521, −3.73228614748983246477352736933, −2.59848668994760664128888018009, −1.19249661233839698683305453004, 0,
1.19249661233839698683305453004, 2.59848668994760664128888018009, 3.73228614748983246477352736933, 4.40501873903252195270502828521, 5.82352281033431510048667037285, 6.59206106647312839868292912255, 7.34994301157373200405754816492, 8.498305487193909611392596711711, 9.220135221196649089483314312911
