L(s) = 1 | − 2-s − 2·3-s − 7·4-s + 16·5-s + 2·6-s − 7·7-s + 15·8-s − 23·9-s − 16·10-s − 8·11-s + 14·12-s + 28·13-s + 7·14-s − 32·15-s + 41·16-s + 54·17-s + 23·18-s − 110·19-s − 112·20-s + 14·21-s + 8·22-s + 48·23-s − 30·24-s + 131·25-s − 28·26-s + 100·27-s + 49·28-s + ⋯ |
L(s) = 1 | − 0.353·2-s − 0.384·3-s − 7/8·4-s + 1.43·5-s + 0.136·6-s − 0.377·7-s + 0.662·8-s − 0.851·9-s − 0.505·10-s − 0.219·11-s + 0.336·12-s + 0.597·13-s + 0.133·14-s − 0.550·15-s + 0.640·16-s + 0.770·17-s + 0.301·18-s − 1.32·19-s − 1.25·20-s + 0.145·21-s + 0.0775·22-s + 0.435·23-s − 0.255·24-s + 1.04·25-s − 0.211·26-s + 0.712·27-s + 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5995661579\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5995661579\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + p T \) |
good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 110 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 110 T + p^{3} T^{2} \) |
| 31 | \( 1 - 12 T + p^{3} T^{2} \) |
| 37 | \( 1 + 246 T + p^{3} T^{2} \) |
| 41 | \( 1 - 182 T + p^{3} T^{2} \) |
| 43 | \( 1 - 128 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 810 T + p^{3} T^{2} \) |
| 61 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 244 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 + 702 T + p^{3} T^{2} \) |
| 79 | \( 1 - 440 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 - 730 T + p^{3} T^{2} \) |
| 97 | \( 1 - 294 T + p^{3} T^{2} \) |
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
−22.29829705980502549266677508752, −21.01874672820152951151166325098, −18.98691076138796378062634315761, −17.66603114438279377377088788569, −16.84418190446002259612227390359, −14.24617763309018446443187607209, −13.00459047543158583812879328378, −10.42619845921982396207673495364, −8.921565108109471505669921413512, −5.74560490678666676322759669793,
5.74560490678666676322759669793, 8.921565108109471505669921413512, 10.42619845921982396207673495364, 13.00459047543158583812879328378, 14.24617763309018446443187607209, 16.84418190446002259612227390359, 17.66603114438279377377088788569, 18.98691076138796378062634315761, 21.01874672820152951151166325098, 22.29829705980502549266677508752