| L(s) = 1 | − 11·3-s − 25·5-s + 49·7-s − 122·9-s − 83·11-s − 83·13-s + 275·15-s − 177·17-s + 2.08e3·19-s − 539·21-s + 3.17e3·23-s + 625·25-s + 4.01e3·27-s − 8.68e3·29-s − 1.63e3·31-s + 913·33-s − 1.22e3·35-s + 4.29e3·37-s + 913·39-s + 2.35e3·41-s − 8.73e3·43-s + 3.05e3·45-s + 3.64e3·47-s + 2.40e3·49-s + 1.94e3·51-s + 3.32e4·53-s + 2.07e3·55-s + ⋯ |
| L(s) = 1 | − 0.705·3-s − 0.447·5-s + 0.377·7-s − 0.502·9-s − 0.206·11-s − 0.136·13-s + 0.315·15-s − 0.148·17-s + 1.32·19-s − 0.266·21-s + 1.24·23-s + 1/5·25-s + 1.05·27-s − 1.91·29-s − 0.305·31-s + 0.145·33-s − 0.169·35-s + 0.516·37-s + 0.0961·39-s + 0.218·41-s − 0.720·43-s + 0.224·45-s + 0.240·47-s + 1/7·49-s + 0.104·51-s + 1.62·53-s + 0.0924·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 11 T + p^{5} T^{2} \) |
| 11 | \( 1 + 83 T + p^{5} T^{2} \) |
| 13 | \( 1 + 83 T + p^{5} T^{2} \) |
| 17 | \( 1 + 177 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2082 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3170 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8681 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1636 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4298 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2356 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8738 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3641 T + p^{5} T^{2} \) |
| 53 | \( 1 - 33268 T + p^{5} T^{2} \) |
| 59 | \( 1 - 30968 T + p^{5} T^{2} \) |
| 61 | \( 1 - 4560 T + p^{5} T^{2} \) |
| 67 | \( 1 + 564 p T + p^{5} T^{2} \) |
| 71 | \( 1 - 59304 T + p^{5} T^{2} \) |
| 73 | \( 1 + 8910 T + p^{5} T^{2} \) |
| 79 | \( 1 + 27589 T + p^{5} T^{2} \) |
| 83 | \( 1 + 67676 T + p^{5} T^{2} \) |
| 89 | \( 1 - 10700 T + p^{5} T^{2} \) |
| 97 | \( 1 - 65075 T + p^{5} T^{2} \) |
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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
−9.529644084356057352292378550857, −8.663438459030118667228494025299, −7.63990485990199462537927448629, −6.90295383577620248334265382466, −5.60026304184630786692771900917, −5.12226615634501502332815133534, −3.83238861987435720654826558375, −2.67432486975539993859751259551, −1.12775448003691978491480585796, 0,
1.12775448003691978491480585796, 2.67432486975539993859751259551, 3.83238861987435720654826558375, 5.12226615634501502332815133534, 5.60026304184630786692771900917, 6.90295383577620248334265382466, 7.63990485990199462537927448629, 8.663438459030118667228494025299, 9.529644084356057352292378550857
