| L(s) = 1 | − 5-s + 2·7-s + 2·17-s + 2·19-s + 25-s + 6·29-s − 2·35-s − 6·37-s − 6·41-s + 2·43-s − 4·47-s − 3·49-s − 2·53-s − 8·59-s − 4·67-s + 12·71-s − 16·73-s − 2·79-s − 8·83-s − 2·85-s + 6·89-s − 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.485·17-s + 0.458·19-s + 1/5·25-s + 1.11·29-s − 0.338·35-s − 0.986·37-s − 0.937·41-s + 0.304·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s − 1.04·59-s − 0.488·67-s + 1.42·71-s − 1.87·73-s − 0.225·79-s − 0.878·83-s − 0.216·85-s + 0.635·89-s − 0.205·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.14746634252477, −13.77434424639344, −13.20460862586871, −12.45825470085235, −12.26307041940894, −11.56284737589125, −11.34983570069579, −10.71830402593594, −10.08494541723699, −9.875955523960149, −8.955725500177749, −8.592915498413002, −8.163790020007114, −7.400469578883132, −7.340276804949492, −6.405270476418017, −6.017169208421624, −5.183107530259076, −4.818568592465656, −4.362002742787056, −3.387682983374988, −3.246103529624219, −2.274841505895402, −1.586159025701159, −0.9618755881788787, 0,
0.9618755881788787, 1.586159025701159, 2.274841505895402, 3.246103529624219, 3.387682983374988, 4.362002742787056, 4.818568592465656, 5.183107530259076, 6.017169208421624, 6.405270476418017, 7.340276804949492, 7.400469578883132, 8.163790020007114, 8.592915498413002, 8.955725500177749, 9.875955523960149, 10.08494541723699, 10.71830402593594, 11.34983570069579, 11.56284737589125, 12.26307041940894, 12.45825470085235, 13.20460862586871, 13.77434424639344, 14.14746634252477
