L(s) = 1 | − 2.12·2-s − 3.47·4-s − 5·5-s + 30.7·7-s + 24.4·8-s + 10.6·10-s − 50.1·11-s − 15.9·13-s − 65.2·14-s − 24.0·16-s − 105.·17-s − 21.3·19-s + 17.3·20-s + 106.·22-s − 136.·23-s + 25·25-s + 33.9·26-s − 106.·28-s + 224.·29-s − 225.·31-s − 144.·32-s + 224.·34-s − 153.·35-s − 416.·37-s + 45.2·38-s − 122.·40-s − 76.1·41-s + ⋯ |
L(s) = 1 | − 0.751·2-s − 0.434·4-s − 0.447·5-s + 1.65·7-s + 1.07·8-s + 0.336·10-s − 1.37·11-s − 0.340·13-s − 1.24·14-s − 0.375·16-s − 1.50·17-s − 0.257·19-s + 0.194·20-s + 1.03·22-s − 1.23·23-s + 0.200·25-s + 0.255·26-s − 0.720·28-s + 1.43·29-s − 1.30·31-s − 0.796·32-s + 1.13·34-s − 0.741·35-s − 1.85·37-s + 0.193·38-s − 0.482·40-s − 0.290·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
good | 2 | \( 1 + 2.12T + 8T^{2} \) |
| 7 | \( 1 - 30.7T + 343T^{2} \) |
| 11 | \( 1 + 50.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 416.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 76.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 31.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 60.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 466.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 95.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 357.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 87.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 412.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 248.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 552.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 291.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 198.T + 9.12e5T^{2} \) |
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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
−12.09537759040453508715304499436, −10.91472828531969028664698990237, −10.34141985996437328778945871911, −8.764726657610715107773695003814, −8.182743296263062209903294619836, −7.29905575078858358403604386712, −5.18630306595890004521382144446, −4.36119619187413278627252558782, −1.97962745682211263790175915637, 0,
1.97962745682211263790175915637, 4.36119619187413278627252558782, 5.18630306595890004521382144446, 7.29905575078858358403604386712, 8.182743296263062209903294619836, 8.764726657610715107773695003814, 10.34141985996437328778945871911, 10.91472828531969028664698990237, 12.09537759040453508715304499436