L(s) = 1 | + 5·5-s + 24.3·7-s − 26.8·11-s − 68.2·13-s + 61.8·17-s + 75.1·19-s − 43.3·23-s + 25·25-s − 174.·29-s − 222.·31-s + 121.·35-s + 67.1·37-s + 22.2·41-s + 84.7·43-s − 585.·47-s + 251.·49-s − 38.3·53-s − 134.·55-s + 92·59-s − 226.·61-s − 341.·65-s − 858.·67-s + 116.·71-s + 911.·73-s − 655.·77-s + 285.·79-s − 999.·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.31·7-s − 0.737·11-s − 1.45·13-s + 0.882·17-s + 0.907·19-s − 0.393·23-s + 0.200·25-s − 1.12·29-s − 1.29·31-s + 0.588·35-s + 0.298·37-s + 0.0845·41-s + 0.300·43-s − 1.81·47-s + 0.734·49-s − 0.0994·53-s − 0.329·55-s + 0.203·59-s − 0.474·61-s − 0.651·65-s − 1.56·67-s + 0.195·71-s + 1.46·73-s − 0.970·77-s + 0.406·79-s − 1.32·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
good | 7 | \( 1 - 24.3T + 343T^{2} \) |
| 11 | \( 1 + 26.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 43.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 67.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 22.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 84.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 585.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 38.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 92T + 2.05e5T^{2} \) |
| 61 | \( 1 + 226.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 858.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 116.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 911.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 285.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 999.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 374.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 227.T + 9.12e5T^{2} \) |
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
−8.031045062360317698133057044264, −7.74344585912297469793807880142, −6.97557442620747893821136293447, −5.58566976342226000246787325249, −5.29094342185355020311232801911, −4.47182675511791707609890332874, −3.23069917940648544562306349341, −2.21502325030531071594558919439, −1.42591659551756577157957946280, 0,
1.42591659551756577157957946280, 2.21502325030531071594558919439, 3.23069917940648544562306349341, 4.47182675511791707609890332874, 5.29094342185355020311232801911, 5.58566976342226000246787325249, 6.97557442620747893821136293447, 7.74344585912297469793807880142, 8.031045062360317698133057044264