| L(s) = 1 | − 5-s − 3·11-s + 6·13-s + 5·17-s + 19-s − 7·23-s − 4·25-s + 2·29-s + 5·31-s − 3·37-s + 2·41-s − 4·43-s + 5·47-s − 53-s + 3·55-s − 15·59-s + 5·61-s − 6·65-s − 9·67-s + 7·73-s − 79-s − 12·83-s − 5·85-s − 7·89-s − 95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.904·11-s + 1.66·13-s + 1.21·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s + 0.371·29-s + 0.898·31-s − 0.493·37-s + 0.312·41-s − 0.609·43-s + 0.729·47-s − 0.137·53-s + 0.404·55-s − 1.95·59-s + 0.640·61-s − 0.744·65-s − 1.09·67-s + 0.819·73-s − 0.112·79-s − 1.31·83-s − 0.542·85-s − 0.741·89-s − 0.102·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−15.62674326691225, −15.17582546011505, −14.16682806987910, −13.92992106035140, −13.50010829696929, −12.73537853281610, −12.29855323126374, −11.67071837612174, −11.31809004282823, −10.47086140713671, −10.24239789598033, −9.608480520751862, −8.787340756131875, −8.267546828437400, −7.840652720746090, −7.406257518249888, −6.424004373672975, −5.959131708900274, −5.487417130685759, −4.644328764599759, −3.957128086376755, −3.430817034620115, −2.778750426799052, −1.794353409647582, −1.043725912036251, 0,
1.043725912036251, 1.794353409647582, 2.778750426799052, 3.430817034620115, 3.957128086376755, 4.644328764599759, 5.487417130685759, 5.959131708900274, 6.424004373672975, 7.406257518249888, 7.840652720746090, 8.267546828437400, 8.787340756131875, 9.608480520751862, 10.24239789598033, 10.47086140713671, 11.31809004282823, 11.67071837612174, 12.29855323126374, 12.73537853281610, 13.50010829696929, 13.92992106035140, 14.16682806987910, 15.17582546011505, 15.62674326691225
