| L(s) = 1 | − 7-s + 4·11-s + 6·13-s + 2·17-s − 6·29-s − 8·31-s + 10·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 2·53-s − 8·59-s − 14·61-s − 12·67-s − 16·71-s − 2·73-s − 4·77-s + 8·79-s − 8·83-s − 10·89-s − 6·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 1.04·59-s − 1.79·61-s − 1.46·67-s − 1.89·71-s − 0.234·73-s − 0.455·77-s + 0.900·79-s − 0.878·83-s − 1.05·89-s − 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−15.67225083859428, −14.94644561349370, −14.67720168082641, −13.97453481354031, −13.49527811742600, −12.93732035748255, −12.51614036892032, −11.72259650091484, −11.30720593768393, −10.87082540907919, −10.18134088651184, −9.453342363461301, −9.046144138277398, −8.666270407425560, −7.687415713700300, −7.426673015063221, −6.404926253783660, −6.147463424058473, −5.633641894516340, −4.641112214827860, −3.958304595354019, −3.520408916186592, −2.849232259202210, −1.603751430783516, −1.289536795897307, 0,
1.289536795897307, 1.603751430783516, 2.849232259202210, 3.520408916186592, 3.958304595354019, 4.641112214827860, 5.633641894516340, 6.147463424058473, 6.404926253783660, 7.426673015063221, 7.687415713700300, 8.666270407425560, 9.046144138277398, 9.453342363461301, 10.18134088651184, 10.87082540907919, 11.30720593768393, 11.72259650091484, 12.51614036892032, 12.93732035748255, 13.49527811742600, 13.97453481354031, 14.67720168082641, 14.94644561349370, 15.67225083859428
