| L(s) = 1 | + 5-s + 4·13-s − 2·17-s + 2·19-s − 8·23-s + 25-s − 4·29-s + 4·31-s − 2·37-s − 12·41-s + 8·43-s − 4·47-s − 7·49-s + 10·53-s + 4·59-s − 6·61-s + 4·65-s − 12·67-s + 16·73-s + 6·79-s + 2·83-s − 2·85-s − 18·89-s + 2·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.10·13-s − 0.485·17-s + 0.458·19-s − 1.66·23-s + 1/5·25-s − 0.742·29-s + 0.718·31-s − 0.328·37-s − 1.87·41-s + 1.21·43-s − 0.583·47-s − 49-s + 1.37·53-s + 0.520·59-s − 0.768·61-s + 0.496·65-s − 1.46·67-s + 1.87·73-s + 0.675·79-s + 0.219·83-s − 0.216·85-s − 1.90·89-s + 0.205·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.89027467666682, −13.71156231267728, −13.34796623669176, −12.68982878022383, −12.13170803951673, −11.71574283672771, −11.16090314710445, −10.67603794299508, −10.15199609417778, −9.684791298775736, −9.205266616614220, −8.496742980615309, −8.265599677025122, −7.613883318448058, −6.905700442497466, −6.487361503256640, −5.851107230538398, −5.569796467812816, −4.762599796300720, −4.205634161583882, −3.576928403168468, −3.088072740712262, −2.149534148020358, −1.763193760896356, −0.9500967486062138, 0,
0.9500967486062138, 1.763193760896356, 2.149534148020358, 3.088072740712262, 3.576928403168468, 4.205634161583882, 4.762599796300720, 5.569796467812816, 5.851107230538398, 6.487361503256640, 6.905700442497466, 7.613883318448058, 8.265599677025122, 8.496742980615309, 9.205266616614220, 9.684791298775736, 10.15199609417778, 10.67603794299508, 11.16090314710445, 11.71574283672771, 12.13170803951673, 12.68982878022383, 13.34796623669176, 13.71156231267728, 13.89027467666682
