L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s + 14-s + 15-s + 16-s − 18-s + 20-s − 21-s + 23-s − 24-s + 27-s − 28-s − 30-s − 31-s − 32-s − 35-s + 36-s − 37-s − 40-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s + 14-s + 15-s + 16-s − 18-s + 20-s − 21-s + 23-s − 24-s + 27-s − 28-s − 30-s − 31-s − 32-s − 35-s + 36-s − 37-s − 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9626499557\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9626499557\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 - T \) |
good | 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−10.08364799642911575077599373671, −9.524083581167165452097143090233, −9.006740162599762706985829499451, −8.116198299088264670089716840466, −7.05834036056947510085191043335, −6.49752987998093054835894794144, −5.36091732647088271884717218480, −3.58716082844993196932502210185, −2.69609415490376309953047836513, −1.64138996354391985831519370642,
1.64138996354391985831519370642, 2.69609415490376309953047836513, 3.58716082844993196932502210185, 5.36091732647088271884717218480, 6.49752987998093054835894794144, 7.05834036056947510085191043335, 8.116198299088264670089716840466, 9.006740162599762706985829499451, 9.524083581167165452097143090233, 10.08364799642911575077599373671