| L(s) = 1 | − 128·2-s − 2.18e3·3-s + 1.63e4·4-s + 2.96e5·5-s + 2.79e5·6-s + 8.23e5·7-s − 2.09e6·8-s + 4.78e6·9-s − 3.79e7·10-s − 7.13e6·11-s − 3.58e7·12-s − 2.36e8·13-s − 1.05e8·14-s − 6.48e8·15-s + 2.68e8·16-s − 2.33e9·17-s − 6.12e8·18-s + 7.02e8·19-s + 4.85e9·20-s − 1.80e9·21-s + 9.12e8·22-s − 1.29e10·23-s + 4.58e9·24-s + 5.73e10·25-s + 3.02e10·26-s − 1.04e10·27-s + 1.34e10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.69·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.19·10-s − 0.110·11-s − 0.288·12-s − 1.04·13-s − 0.267·14-s − 0.979·15-s + 1/4·16-s − 1.38·17-s − 0.235·18-s + 0.180·19-s + 0.848·20-s − 0.218·21-s + 0.0780·22-s − 0.793·23-s + 0.204·24-s + 1.87·25-s + 0.739·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{7} T \) |
| 3 | \( 1 + p^{7} T \) |
| 7 | \( 1 - p^{7} T \) |
| good | 5 | \( 1 - 296442 T + p^{15} T^{2} \) |
| 11 | \( 1 + 648288 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 18205834 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 2335326882 T + p^{15} T^{2} \) |
| 19 | \( 1 - 702711596 T + p^{15} T^{2} \) |
| 23 | \( 1 + 12958741404 T + p^{15} T^{2} \) |
| 29 | \( 1 - 66702541614 T + p^{15} T^{2} \) |
| 31 | \( 1 + 29492707192 T + p^{15} T^{2} \) |
| 37 | \( 1 + 935632269682 T + p^{15} T^{2} \) |
| 41 | \( 1 + 1076568776178 T + p^{15} T^{2} \) |
| 43 | \( 1 - 2788495492052 T + p^{15} T^{2} \) |
| 47 | \( 1 + 4151951670984 T + p^{15} T^{2} \) |
| 53 | \( 1 - 8187896340318 T + p^{15} T^{2} \) |
| 59 | \( 1 + 17963955588252 T + p^{15} T^{2} \) |
| 61 | \( 1 - 14067282247670 T + p^{15} T^{2} \) |
| 67 | \( 1 + 91494354339484 T + p^{15} T^{2} \) |
| 71 | \( 1 - 85920788160684 T + p^{15} T^{2} \) |
| 73 | \( 1 + 123132339461662 T + p^{15} T^{2} \) |
| 79 | \( 1 + 280698219508408 T + p^{15} T^{2} \) |
| 83 | \( 1 + 45613464690684 T + p^{15} T^{2} \) |
| 89 | \( 1 - 102721846037550 T + p^{15} T^{2} \) |
| 97 | \( 1 - 1062760278868490 T + p^{15} T^{2} \) |
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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
−12.03648812775172499705898340459, −10.65410658860210706812701801692, −9.886061754101103176366129684298, −8.801823299085123796354736920847, −7.08445647294083872096541423873, −6.00764010405491696130859001880, −4.87234036619838256719386238068, −2.42931517165284498834443279297, −1.55369348891294452152881237507, 0,
1.55369348891294452152881237507, 2.42931517165284498834443279297, 4.87234036619838256719386238068, 6.00764010405491696130859001880, 7.08445647294083872096541423873, 8.801823299085123796354736920847, 9.886061754101103176366129684298, 10.65410658860210706812701801692, 12.03648812775172499705898340459
