| L(s) = 1 | + 5-s + 7-s − 4·11-s − 2·13-s + 6·17-s − 8·23-s + 25-s − 10·29-s + 8·31-s + 35-s + 2·37-s + 2·41-s − 8·43-s + 4·47-s + 49-s − 10·53-s − 4·55-s + 4·59-s − 6·61-s − 2·65-s − 12·71-s − 6·73-s − 4·77-s + 8·79-s − 4·83-s + 6·85-s − 14·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.66·23-s + 1/5·25-s − 1.85·29-s + 1.43·31-s + 0.169·35-s + 0.328·37-s + 0.312·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 1.37·53-s − 0.539·55-s + 0.520·59-s − 0.768·61-s − 0.248·65-s − 1.42·71-s − 0.702·73-s − 0.455·77-s + 0.900·79-s − 0.439·83-s + 0.650·85-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.79135801031302271266409954859, −7.44352128739462254252010249585, −6.26827399418199187607160496703, −5.64653758882322978280272441138, −5.08101436197765081520038676197, −4.20702332328698474598070130580, −3.18942031728552476894395267386, −2.37350761210594341027206433932, −1.47735506679789706444720092664, 0,
1.47735506679789706444720092664, 2.37350761210594341027206433932, 3.18942031728552476894395267386, 4.20702332328698474598070130580, 5.08101436197765081520038676197, 5.64653758882322978280272441138, 6.26827399418199187607160496703, 7.44352128739462254252010249585, 7.79135801031302271266409954859
