| L(s) = 1 | − 64·2-s + 1.83e3·3-s + 4.09e3·4-s − 3.99e3·5-s − 1.17e5·6-s − 2.62e5·8-s + 1.77e6·9-s + 2.55e5·10-s + 1.61e6·11-s + 7.52e6·12-s + 1.08e7·13-s − 7.32e6·15-s + 1.67e7·16-s − 6.05e7·17-s − 1.13e8·18-s + 2.43e8·19-s − 1.63e7·20-s − 1.03e8·22-s − 6.06e8·23-s − 4.81e8·24-s − 1.20e9·25-s − 6.96e8·26-s + 3.34e8·27-s + 5.25e9·29-s + 4.68e8·30-s + 1.82e9·31-s − 1.07e9·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.45·3-s + 1/2·4-s − 0.114·5-s − 1.02·6-s − 0.353·8-s + 1.11·9-s + 0.0807·10-s + 0.275·11-s + 0.727·12-s + 0.625·13-s − 0.166·15-s + 1/4·16-s − 0.608·17-s − 0.787·18-s + 1.18·19-s − 0.0571·20-s − 0.194·22-s − 0.853·23-s − 0.514·24-s − 0.986·25-s − 0.441·26-s + 0.166·27-s + 1.64·29-s + 0.117·30-s + 0.369·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.923926404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923926404\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{6} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 68 p^{3} T + p^{13} T^{2} \) |
| 5 | \( 1 + 798 p T + p^{13} T^{2} \) |
| 11 | \( 1 - 147252 p T + p^{13} T^{2} \) |
| 13 | \( 1 - 10878466 T + p^{13} T^{2} \) |
| 17 | \( 1 + 60569298 T + p^{13} T^{2} \) |
| 19 | \( 1 - 243131740 T + p^{13} T^{2} \) |
| 23 | \( 1 + 606096456 T + p^{13} T^{2} \) |
| 29 | \( 1 - 181332390 p T + p^{13} T^{2} \) |
| 31 | \( 1 - 1824312928 T + p^{13} T^{2} \) |
| 37 | \( 1 + 3005875402 T + p^{13} T^{2} \) |
| 41 | \( 1 - 49704880758 T + p^{13} T^{2} \) |
| 43 | \( 1 - 58766693084 T + p^{13} T^{2} \) |
| 47 | \( 1 - 42095878032 T + p^{13} T^{2} \) |
| 53 | \( 1 + 181140755706 T + p^{13} T^{2} \) |
| 59 | \( 1 + 206730587820 T + p^{13} T^{2} \) |
| 61 | \( 1 - 124479015058 T + p^{13} T^{2} \) |
| 67 | \( 1 - 95665133588 T + p^{13} T^{2} \) |
| 71 | \( 1 + 371436487128 T + p^{13} T^{2} \) |
| 73 | \( 1 - 1800576064726 T + p^{13} T^{2} \) |
| 79 | \( 1 - 1557932091920 T + p^{13} T^{2} \) |
| 83 | \( 1 + 2492790917604 T + p^{13} T^{2} \) |
| 89 | \( 1 + 2994235754490 T + p^{13} T^{2} \) |
| 97 | \( 1 + 4382492665058 T + p^{13} T^{2} \) |
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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
−11.19255271270105010403500040183, −9.900498910442252898640593181504, −9.119076270574065202773432487165, −8.229212027505187691696299321850, −7.46236367997120446482240077276, −6.10128015004342119747304897120, −4.17801696528487255221298290051, −3.06911598468550641034433190975, −2.06907767088645339598656569221, −0.865709798055600662438747831136,
0.865709798055600662438747831136, 2.06907767088645339598656569221, 3.06911598468550641034433190975, 4.17801696528487255221298290051, 6.10128015004342119747304897120, 7.46236367997120446482240077276, 8.229212027505187691696299321850, 9.119076270574065202773432487165, 9.900498910442252898640593181504, 11.19255271270105010403500040183
