L(s) = 1 | − 1.41i·5-s + 1.41·7-s + 2i·11-s − 2·17-s + 4i·19-s − 2.82·23-s + 2.99·25-s − 9.89i·29-s + 7.07·31-s − 2.00i·35-s + 8.48i·37-s + 6·41-s − 8i·43-s + 2.82·47-s − 5·49-s + ⋯ |
L(s) = 1 | − 0.632i·5-s + 0.534·7-s + 0.603i·11-s − 0.485·17-s + 0.917i·19-s − 0.589·23-s + 0.599·25-s − 1.83i·29-s + 1.27·31-s − 0.338i·35-s + 1.39i·37-s + 0.937·41-s − 1.21i·43-s + 0.412·47-s − 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.013607158\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.013607158\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 9.89iT - 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 1.41iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 - 14.1iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.268535267744260627670584311593, −7.78423797141011679171442599603, −6.87583827286314315173794216136, −6.08474469930001689170614329683, −5.34979043634966811847091014094, −4.44413664508296496714480593828, −4.11395722423283056664546112277, −2.75109823258849035998954484800, −1.88860240265936070572036459358, −0.854833749996929003678924908984,
0.75062795931874660358265460023, 2.03067494994519599869099220131, 2.89722773300279085305839051145, 3.67143712359787460018648099296, 4.71644448163048735560586080506, 5.25175197735978266990159926893, 6.40880769833428231566551980036, 6.67826571613589883126997453534, 7.68472034845463256759112926822, 8.221894519699595911132818334118