| L(s) = 1 | − 5-s − 7-s − 6·11-s − 2·13-s + 3·17-s + 6·19-s + 23-s + 25-s − 3·29-s + 3·31-s + 35-s + 37-s − 9·41-s + 8·43-s + 4·47-s − 6·49-s − 53-s + 6·55-s + 59-s + 8·61-s + 2·65-s + 7·67-s − 5·71-s − 6·73-s + 6·77-s − 11·83-s − 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.80·11-s − 0.554·13-s + 0.727·17-s + 1.37·19-s + 0.208·23-s + 1/5·25-s − 0.557·29-s + 0.538·31-s + 0.169·35-s + 0.164·37-s − 1.40·41-s + 1.21·43-s + 0.583·47-s − 6/7·49-s − 0.137·53-s + 0.809·55-s + 0.130·59-s + 1.02·61-s + 0.248·65-s + 0.855·67-s − 0.593·71-s − 0.702·73-s + 0.683·77-s − 1.20·83-s − 0.325·85-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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−16.03432504471823, −15.73844457630931, −15.23656623363786, −14.54836159653310, −13.96555352661478, −13.40049818347410, −12.74392368969990, −12.50087256186474, −11.59120859291971, −11.34854066636955, −10.35006986209096, −10.12206502036642, −9.533329364873269, −8.723503700513909, −8.072974631870865, −7.470797067502147, −7.263519309640733, −6.258909846176002, −5.413653848931338, −5.181701176002095, −4.347470867908775, −3.358112300532148, −2.963775013290966, −2.171352920627751, −0.9503867477407425, 0,
0.9503867477407425, 2.171352920627751, 2.963775013290966, 3.358112300532148, 4.347470867908775, 5.181701176002095, 5.413653848931338, 6.258909846176002, 7.263519309640733, 7.470797067502147, 8.072974631870865, 8.723503700513909, 9.533329364873269, 10.12206502036642, 10.35006986209096, 11.34854066636955, 11.59120859291971, 12.50087256186474, 12.74392368969990, 13.40049818347410, 13.96555352661478, 14.54836159653310, 15.23656623363786, 15.73844457630931, 16.03432504471823
