L(s) = 1 | − 2·5-s − 11-s − 6·13-s − 2·17-s + 8·19-s − 4·23-s − 25-s + 2·29-s − 8·31-s − 6·37-s − 2·41-s + 8·43-s + 4·47-s + 2·53-s + 2·55-s − 12·59-s + 10·61-s + 12·65-s − 12·67-s − 12·71-s − 10·73-s + 8·79-s + 12·83-s + 4·85-s + 10·89-s − 16·95-s + 14·97-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.301·11-s − 1.66·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.312·41-s + 1.21·43-s + 0.583·47-s + 0.274·53-s + 0.269·55-s − 1.56·59-s + 1.28·61-s + 1.48·65-s − 1.46·67-s − 1.42·71-s − 1.17·73-s + 0.900·79-s + 1.31·83-s + 0.433·85-s + 1.05·89-s − 1.64·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5343149633\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5343149633\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.37916782055829, −12.14982838339049, −11.88869146419793, −11.41435210952895, −10.83600655551141, −10.32436443662922, −10.00808198334878, −9.322977447589874, −9.118411127994383, −8.505242277884166, −7.739631265085301, −7.566238087948747, −7.306487960317827, −6.776106925545397, −5.862769705555487, −5.673765997267915, −4.887658923466727, −4.678891292929921, −4.031621314791035, −3.393505185503608, −3.094779356659490, −2.275266654614972, −1.914486965118006, −0.9741743811477432, −0.2178492054879602,
0.2178492054879602, 0.9741743811477432, 1.914486965118006, 2.275266654614972, 3.094779356659490, 3.393505185503608, 4.031621314791035, 4.678891292929921, 4.887658923466727, 5.673765997267915, 5.862769705555487, 6.776106925545397, 7.306487960317827, 7.566238087948747, 7.739631265085301, 8.505242277884166, 9.118411127994383, 9.322977447589874, 10.00808198334878, 10.32436443662922, 10.83600655551141, 11.41435210952895, 11.88869146419793, 12.14982838339049, 12.37916782055829