| L(s) = 1 | + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s + 13-s + 4·14-s + 16-s + 2·17-s − 4·19-s + 20-s + 25-s + 26-s + 4·28-s + 2·29-s + 8·31-s + 32-s + 2·34-s + 4·35-s − 6·37-s − 4·38-s + 40-s − 6·41-s + 4·43-s + 9·49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.196·26-s + 0.755·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.676·35-s − 0.986·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.142886197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.142886197\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.45569222215774, −13.07748239239386, −12.26488159452868, −12.08431208150216, −11.58989841556871, −11.00634927376762, −10.62793830955341, −10.21518576716579, −9.660990348414562, −8.871570867384282, −8.387291957419592, −8.172210950454605, −7.446030794527600, −6.976264736807526, −6.320020510165017, −5.956365997840933, −5.207244954752711, −4.962476396563613, −4.405338765021646, −3.858889563823052, −3.191577128719314, −2.449211216834150, −1.979340277078878, −1.391741982821249, −0.7139989702083412,
0.7139989702083412, 1.391741982821249, 1.979340277078878, 2.449211216834150, 3.191577128719314, 3.858889563823052, 4.405338765021646, 4.962476396563613, 5.207244954752711, 5.956365997840933, 6.320020510165017, 6.976264736807526, 7.446030794527600, 8.172210950454605, 8.387291957419592, 8.871570867384282, 9.660990348414562, 10.21518576716579, 10.62793830955341, 11.00634927376762, 11.58989841556871, 12.08431208150216, 12.26488159452868, 13.07748239239386, 13.45569222215774
