| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 7-s − 2·9-s + 3·11-s − 2·12-s + 13-s + 2·14-s − 4·16-s − 4·18-s − 21-s + 6·22-s − 6·23-s + 2·26-s + 5·27-s + 2·28-s + 5·29-s − 2·31-s − 8·32-s − 3·33-s − 4·36-s − 2·37-s − 39-s − 2·41-s − 2·42-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s + 0.277·13-s + 0.534·14-s − 16-s − 0.942·18-s − 0.218·21-s + 1.27·22-s − 1.25·23-s + 0.392·26-s + 0.962·27-s + 0.377·28-s + 0.928·29-s − 0.359·31-s − 1.41·32-s − 0.522·33-s − 2/3·36-s − 0.328·37-s − 0.160·39-s − 0.312·41-s − 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.64002564125645, −14.20978926514614, −13.81394253146794, −13.33426010054795, −12.67902887038101, −12.13225818568645, −11.80127670845848, −11.47035424197259, −10.93528669008964, −10.26341222699785, −9.693540103108074, −8.904963343182625, −8.491099265509478, −7.946341807422589, −6.913490484003947, −6.647273098803020, −6.069066533562253, −5.515894966802146, −5.144459889976130, −4.452661876647726, −3.893401299926782, −3.439176202902306, −2.616411801385783, −1.987692128314599, −1.031503631318047, 0,
1.031503631318047, 1.987692128314599, 2.616411801385783, 3.439176202902306, 3.893401299926782, 4.452661876647726, 5.144459889976130, 5.515894966802146, 6.069066533562253, 6.647273098803020, 6.913490484003947, 7.946341807422589, 8.491099265509478, 8.904963343182625, 9.693540103108074, 10.26341222699785, 10.93528669008964, 11.47035424197259, 11.80127670845848, 12.13225818568645, 12.67902887038101, 13.33426010054795, 13.81394253146794, 14.20978926514614, 14.64002564125645
