| L(s) = 1 | − 256·2-s + 6.56e3·3-s + 6.55e4·4-s + 3.90e5·5-s − 1.67e6·6-s + 1.92e6·7-s − 1.67e7·8-s + 4.30e7·9-s − 1.00e8·10-s + 3.52e8·11-s + 4.29e8·12-s − 3.50e9·13-s − 4.93e8·14-s + 2.56e9·15-s + 4.29e9·16-s − 5.49e10·17-s − 1.10e10·18-s + 3.10e10·19-s + 2.56e10·20-s + 1.26e10·21-s − 9.03e10·22-s − 3.16e11·23-s − 1.10e11·24-s + 1.52e11·25-s + 8.96e11·26-s + 2.82e11·27-s + 1.26e11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.126·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.496·11-s + 0.288·12-s − 1.19·13-s − 0.0894·14-s + 0.258·15-s + 1/4·16-s − 1.91·17-s − 0.235·18-s + 0.420·19-s + 0.223·20-s + 0.0730·21-s − 0.351·22-s − 0.842·23-s − 0.204·24-s + 1/5·25-s + 0.841·26-s + 0.192·27-s + 0.0632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{8} T \) |
| 3 | \( 1 - p^{8} T \) |
| 5 | \( 1 - p^{8} T \) |
| good | 7 | \( 1 - 275648 p T + p^{17} T^{2} \) |
| 11 | \( 1 - 32089620 p T + p^{17} T^{2} \) |
| 13 | \( 1 + 269230786 p T + p^{17} T^{2} \) |
| 17 | \( 1 + 54970977894 T + p^{17} T^{2} \) |
| 19 | \( 1 - 31092978236 T + p^{17} T^{2} \) |
| 23 | \( 1 + 316297396800 T + p^{17} T^{2} \) |
| 29 | \( 1 - 3237919791294 T + p^{17} T^{2} \) |
| 31 | \( 1 + 6515774619112 T + p^{17} T^{2} \) |
| 37 | \( 1 - 26233622989982 T + p^{17} T^{2} \) |
| 41 | \( 1 - 44280220470714 T + p^{17} T^{2} \) |
| 43 | \( 1 + 6124629403444 T + p^{17} T^{2} \) |
| 47 | \( 1 + 44171951954040 T + p^{17} T^{2} \) |
| 53 | \( 1 + 489680642744826 T + p^{17} T^{2} \) |
| 59 | \( 1 + 820004055924180 T + p^{17} T^{2} \) |
| 61 | \( 1 + 2460927715419250 T + p^{17} T^{2} \) |
| 67 | \( 1 + 4871518631786044 T + p^{17} T^{2} \) |
| 71 | \( 1 - 4555031256305160 T + p^{17} T^{2} \) |
| 73 | \( 1 - 1598431505471162 T + p^{17} T^{2} \) |
| 79 | \( 1 + 7625872763936872 T + p^{17} T^{2} \) |
| 83 | \( 1 + 19662256866055308 T + p^{17} T^{2} \) |
| 89 | \( 1 + 15591056930648502 T + p^{17} T^{2} \) |
| 97 | \( 1 + 17776788467854078 T + p^{17} T^{2} \) |
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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
−12.56753771392665209355136046755, −11.15070717436039773585406765604, −9.786905243774846076752446073283, −8.951312293997072883047531184600, −7.61042655188708599444976505881, −6.37952054659050575999520018745, −4.52251211802375842447117939700, −2.69857510886641711976189221373, −1.66479805116529233065018917827, 0,
1.66479805116529233065018917827, 2.69857510886641711976189221373, 4.52251211802375842447117939700, 6.37952054659050575999520018745, 7.61042655188708599444976505881, 8.951312293997072883047531184600, 9.786905243774846076752446073283, 11.15070717436039773585406765604, 12.56753771392665209355136046755
