| L(s) = 1 | − 5-s + 7-s + 13-s + 8·17-s − 5·19-s − 4·25-s − 5·29-s + 6·31-s − 35-s − 11·37-s − 8·43-s + 7·47-s + 49-s − 2·53-s + 7·59-s − 2·61-s − 65-s + 7·67-s + 7·73-s − 2·79-s − 12·83-s − 8·85-s − 2·89-s + 91-s + 5·95-s − 8·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.277·13-s + 1.94·17-s − 1.14·19-s − 4/5·25-s − 0.928·29-s + 1.07·31-s − 0.169·35-s − 1.80·37-s − 1.21·43-s + 1.02·47-s + 1/7·49-s − 0.274·53-s + 0.911·59-s − 0.256·61-s − 0.124·65-s + 0.855·67-s + 0.819·73-s − 0.225·79-s − 1.31·83-s − 0.867·85-s − 0.211·89-s + 0.104·91-s + 0.512·95-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.94955801320517, −13.30682593876343, −12.73200995654003, −12.29317520674753, −11.89772907033388, −11.42650332641794, −10.94958904372059, −10.26202436145595, −10.06892745176725, −9.469045613205271, −8.726668446919934, −8.303794163082229, −8.028398336514291, −7.297749473867623, −7.019384867208512, −6.189607863566357, −5.775531903290788, −5.208490210550197, −4.698281687335391, −3.900156221036673, −3.640428651559676, −2.999481854309436, −2.149439019242962, −1.609280068112317, −0.8561730805353134, 0,
0.8561730805353134, 1.609280068112317, 2.149439019242962, 2.999481854309436, 3.640428651559676, 3.900156221036673, 4.698281687335391, 5.208490210550197, 5.775531903290788, 6.189607863566357, 7.019384867208512, 7.297749473867623, 8.028398336514291, 8.303794163082229, 8.726668446919934, 9.469045613205271, 10.06892745176725, 10.26202436145595, 10.94958904372059, 11.42650332641794, 11.89772907033388, 12.29317520674753, 12.73200995654003, 13.30682593876343, 13.94955801320517
