| L(s) = 1 | + 216·2-s − 3.34e3·3-s + 1.38e4·4-s + 5.21e4·5-s − 7.23e5·6-s + 2.82e6·7-s − 4.07e6·8-s − 3.13e6·9-s + 1.12e7·10-s + 2.05e7·11-s − 4.64e7·12-s − 1.90e8·13-s + 6.09e8·14-s − 1.74e8·15-s − 1.33e9·16-s + 1.64e9·17-s − 6.78e8·18-s + 1.56e9·19-s + 7.23e8·20-s − 9.44e9·21-s + 4.44e9·22-s + 9.45e9·23-s + 1.36e10·24-s − 2.78e10·25-s − 4.10e10·26-s + 5.85e10·27-s + 3.91e10·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.883·3-s + 0.423·4-s + 0.298·5-s − 1.05·6-s + 1.29·7-s − 0.687·8-s − 0.218·9-s + 0.355·10-s + 0.318·11-s − 0.374·12-s − 0.840·13-s + 1.54·14-s − 0.263·15-s − 1.24·16-s + 0.973·17-s − 0.261·18-s + 0.401·19-s + 0.126·20-s − 1.14·21-s + 0.380·22-s + 0.578·23-s + 0.607·24-s − 0.911·25-s − 1.00·26-s + 1.07·27-s + 0.549·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+15/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.520561669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520561669\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 27 p^{3} T + p^{15} T^{2} \) |
| 3 | \( 1 + 124 p^{3} T + p^{15} T^{2} \) |
| 5 | \( 1 - 10422 p T + p^{15} T^{2} \) |
| 7 | \( 1 - 403208 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 1871532 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 14621026 p T + p^{15} T^{2} \) |
| 17 | \( 1 - 1646527986 T + p^{15} T^{2} \) |
| 19 | \( 1 - 1563257180 T + p^{15} T^{2} \) |
| 23 | \( 1 - 9451116072 T + p^{15} T^{2} \) |
| 29 | \( 1 + 36902568330 T + p^{15} T^{2} \) |
| 31 | \( 1 - 71588483552 T + p^{15} T^{2} \) |
| 37 | \( 1 + 1033652081554 T + p^{15} T^{2} \) |
| 41 | \( 1 - 1641974018202 T + p^{15} T^{2} \) |
| 43 | \( 1 + 492403109308 T + p^{15} T^{2} \) |
| 47 | \( 1 + 3410684952624 T + p^{15} T^{2} \) |
| 53 | \( 1 - 6797151655902 T + p^{15} T^{2} \) |
| 59 | \( 1 - 167099268060 p T + p^{15} T^{2} \) |
| 61 | \( 1 - 4931842626902 T + p^{15} T^{2} \) |
| 67 | \( 1 + 28837826625364 T + p^{15} T^{2} \) |
| 71 | \( 1 - 125050114914552 T + p^{15} T^{2} \) |
| 73 | \( 1 + 82171455513478 T + p^{15} T^{2} \) |
| 79 | \( 1 + 25413078694480 T + p^{15} T^{2} \) |
| 83 | \( 1 + 281736730890468 T + p^{15} T^{2} \) |
| 89 | \( 1 - 715618564776810 T + p^{15} T^{2} \) |
| 97 | \( 1 - 612786136081826 T + p^{15} T^{2} \) |
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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
−31.31083306980598975354369589119, −29.69484151629079972214366719026, −27.56748408646213265713341304974, −24.35638841200895818548143833510, −22.83359228719678482192637867477, −21.26112311281979572877543178918, −17.51673159287656967480964579195, −14.40110767072392844184488566452, −11.82395546014304725015828959152, −5.26502022758204893749954155127,
5.26502022758204893749954155127, 11.82395546014304725015828959152, 14.40110767072392844184488566452, 17.51673159287656967480964579195, 21.26112311281979572877543178918, 22.83359228719678482192637867477, 24.35638841200895818548143833510, 27.56748408646213265713341304974, 29.69484151629079972214366719026, 31.31083306980598975354369589119
