| L(s) = 1 | + 5-s + 2·13-s + 2·17-s − 8·23-s + 25-s − 2·29-s + 8·31-s + 6·37-s − 2·41-s + 43-s − 7·49-s + 2·53-s + 8·59-s + 10·61-s + 2·65-s − 4·67-s + 14·73-s − 8·79-s − 4·83-s + 2·85-s + 6·89-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.554·13-s + 0.485·17-s − 1.66·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.986·37-s − 0.312·41-s + 0.152·43-s − 49-s + 0.274·53-s + 1.04·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s + 1.63·73-s − 0.900·79-s − 0.439·83-s + 0.216·85-s + 0.635·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.69395752664985, −13.38397835501464, −12.87093554110639, −12.29995535063400, −11.86005532470587, −11.38523803964598, −10.92045677148299, −10.18654403046272, −9.952035303522704, −9.551104292952304, −8.839775644416598, −8.323772559915073, −7.952280686930164, −7.443591283575711, −6.561607309668946, −6.402685814139969, −5.758849696934400, −5.294130632410781, −4.671417533597517, −3.929191221175111, −3.675976866170140, −2.710794760759903, −2.356993470341955, −1.518116899211581, −0.9713875288285294, 0,
0.9713875288285294, 1.518116899211581, 2.356993470341955, 2.710794760759903, 3.675976866170140, 3.929191221175111, 4.671417533597517, 5.294130632410781, 5.758849696934400, 6.402685814139969, 6.561607309668946, 7.443591283575711, 7.952280686930164, 8.323772559915073, 8.839775644416598, 9.551104292952304, 9.952035303522704, 10.18654403046272, 10.92045677148299, 11.38523803964598, 11.86005532470587, 12.29995535063400, 12.87093554110639, 13.38397835501464, 13.69395752664985
