L(s) = 1 | − 2-s + 2·3-s + 4-s − 3·5-s − 2·6-s − 8-s + 9-s + 3·10-s + 6·11-s + 2·12-s + 13-s − 6·15-s + 16-s − 6·17-s − 18-s − 2·19-s − 3·20-s − 6·22-s − 2·24-s + 4·25-s − 26-s − 4·27-s − 9·29-s + 6·30-s + 4·31-s − 32-s + 12·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 1.34·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.80·11-s + 0.577·12-s + 0.277·13-s − 1.54·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.670·20-s − 1.27·22-s − 0.408·24-s + 4/5·25-s − 0.196·26-s − 0.769·27-s − 1.67·29-s + 1.09·30-s + 0.718·31-s − 0.176·32-s + 2.08·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.86692397723530, −14.39491373551152, −13.85916416091419, −13.16641608755470, −12.75624028099747, −11.94626705188856, −11.58577042354898, −11.17027824659124, −10.79559738624569, −9.760236061327746, −9.375224023945534, −8.920528616867979, −8.498645513557648, −8.126706640474729, −7.418433309756591, −7.105888777087304, −6.364201456476319, −5.967264929824126, −4.687734960274530, −4.215267565637384, −3.632176534094180, −3.330018072277435, −2.315569237628697, −1.860770220471151, −0.9050240765993498, 0,
0.9050240765993498, 1.860770220471151, 2.315569237628697, 3.330018072277435, 3.632176534094180, 4.215267565637384, 4.687734960274530, 5.967264929824126, 6.364201456476319, 7.105888777087304, 7.418433309756591, 8.126706640474729, 8.498645513557648, 8.920528616867979, 9.375224023945534, 9.760236061327746, 10.79559738624569, 11.17027824659124, 11.58577042354898, 11.94626705188856, 12.75624028099747, 13.16641608755470, 13.85916416091419, 14.39491373551152, 14.86692397723530