| L(s) = 1 | − 5-s + 2·7-s − 4·11-s + 13-s + 4·19-s − 23-s + 25-s + 7·29-s + 7·31-s − 2·35-s − 4·37-s − 3·41-s − 6·43-s − 13·47-s − 3·49-s − 10·53-s + 4·55-s − 8·59-s − 65-s − 8·67-s + 13·71-s + 11·73-s − 8·77-s − 4·79-s − 4·83-s + 6·89-s + 2·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 1.20·11-s + 0.277·13-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 1.29·29-s + 1.25·31-s − 0.338·35-s − 0.657·37-s − 0.468·41-s − 0.914·43-s − 1.89·47-s − 3/7·49-s − 1.37·53-s + 0.539·55-s − 1.04·59-s − 0.124·65-s − 0.977·67-s + 1.54·71-s + 1.28·73-s − 0.911·77-s − 0.450·79-s − 0.439·83-s + 0.635·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.00856076373689, −15.65700083607234, −15.26584295901798, −14.48166090939636, −13.98589821127550, −13.49957055518529, −12.90087169521539, −12.14485043432741, −11.81455079940254, −11.11363840503801, −10.68725683187927, −9.986519679362308, −9.540068310951588, −8.504842798860716, −8.097984770411042, −7.863483318450232, −6.922340398689295, −6.397953544985723, −5.501295539785243, −4.814884749140392, −4.609615905428687, −3.363248311201318, −3.007370352213119, −1.971802105009008, −1.133621613720938, 0,
1.133621613720938, 1.971802105009008, 3.007370352213119, 3.363248311201318, 4.609615905428687, 4.814884749140392, 5.501295539785243, 6.397953544985723, 6.922340398689295, 7.863483318450232, 8.097984770411042, 8.504842798860716, 9.540068310951588, 9.986519679362308, 10.68725683187927, 11.11363840503801, 11.81455079940254, 12.14485043432741, 12.90087169521539, 13.49957055518529, 13.98589821127550, 14.48166090939636, 15.26584295901798, 15.65700083607234, 16.00856076373689
