L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s + 2·7-s − 8-s + 9-s + 2·10-s − 3·11-s − 12-s + 3·13-s − 2·14-s + 2·15-s + 16-s − 5·17-s − 18-s + 19-s − 2·20-s − 2·21-s + 3·22-s + 4·23-s + 24-s − 25-s − 3·26-s − 27-s + 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.904·11-s − 0.288·12-s + 0.832·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.229·19-s − 0.447·20-s − 0.436·21-s + 0.639·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.588·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−10.10212646936986201287332253082, −8.662452261962971816691461028795, −8.367059456801909353888896766196, −7.31589664936853764163787942878, −6.62339846201472136021011440428, −5.35622838853877625371142570288, −4.51306938620050337716148418007, −3.21001886620242484631764818646, −1.63085640210764186035564524216, 0,
1.63085640210764186035564524216, 3.21001886620242484631764818646, 4.51306938620050337716148418007, 5.35622838853877625371142570288, 6.62339846201472136021011440428, 7.31589664936853764163787942878, 8.367059456801909353888896766196, 8.662452261962971816691461028795, 10.10212646936986201287332253082