L(s) = 1 | − 0.709·2-s − 1.77·3-s − 0.497·4-s + 5-s + 1.25·6-s + 0.241·7-s + 1.06·8-s + 2.13·9-s − 0.709·10-s + 1.94·11-s + 0.880·12-s − 13-s − 0.170·14-s − 1.77·15-s − 0.255·16-s + 1.49·17-s − 1.51·18-s − 0.497·20-s − 0.426·21-s − 1.37·22-s − 1.13·23-s − 1.88·24-s + 25-s + 0.709·26-s − 2.01·27-s − 0.119·28-s + 1.25·30-s + ⋯ |
L(s) = 1 | − 0.709·2-s − 1.77·3-s − 0.497·4-s + 5-s + 1.25·6-s + 0.241·7-s + 1.06·8-s + 2.13·9-s − 0.709·10-s + 1.94·11-s + 0.880·12-s − 13-s − 0.170·14-s − 1.77·15-s − 0.255·16-s + 1.49·17-s − 1.51·18-s − 0.497·20-s − 0.426·21-s − 1.37·22-s − 1.13·23-s − 1.88·24-s + 25-s + 0.709·26-s − 2.01·27-s − 0.119·28-s + 1.25·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5421247023\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5421247023\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 0.709T + T^{2} \) |
| 3 | \( 1 + 1.77T + T^{2} \) |
| 7 | \( 1 - 0.241T + T^{2} \) |
| 11 | \( 1 - 1.94T + T^{2} \) |
| 17 | \( 1 - 1.49T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.13T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.709T + T^{2} \) |
| 47 | \( 1 - 1.13T + T^{2} \) |
| 53 | \( 1 - 1.94T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.94T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.241T + T^{2} \) |
| 97 | \( 1 - 1.77T + T^{2} \) |
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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
−9.528874280167046422022864038961, −8.884948165577991964994463960330, −7.58316067493780853719678788404, −6.97665717766408960704612331112, −6.00638195981166388042531833039, −5.54184251270767931524445381287, −4.67279901674174423321318360159, −3.88693579701816528991417045468, −1.77262720499890806050144658160, −0.976066662750734410475717357659,
0.976066662750734410475717357659, 1.77262720499890806050144658160, 3.88693579701816528991417045468, 4.67279901674174423321318360159, 5.54184251270767931524445381287, 6.00638195981166388042531833039, 6.97665717766408960704612331112, 7.58316067493780853719678788404, 8.884948165577991964994463960330, 9.528874280167046422022864038961