L(s) = 1 | − i·3-s − i·7-s − 9-s − 3·11-s + i·13-s + 3i·17-s + 4·19-s − 21-s + i·27-s + 29-s − 4·31-s + 3i·33-s − 4i·37-s + 39-s − 6·41-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.377i·7-s − 0.333·9-s − 0.904·11-s + 0.277i·13-s + 0.727i·17-s + 0.917·19-s − 0.218·21-s + 0.192i·27-s + 0.185·29-s − 0.718·31-s + 0.522i·33-s − 0.657i·37-s + 0.160·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.642579263\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.642579263\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 4iT - 97T^{2} \) |
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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
−7.73700876448800588184632957105, −7.01462808770279283332720762477, −6.52366468739213696186012979243, −5.51925938510397528826895259003, −5.18338579415761739295889701731, −4.08931230960509471221074270564, −3.40495284795529678081957291284, −2.48573641389579723535046864306, −1.68436290022261036306812622770, −0.63737799036113212848496097215,
0.59476717478288962104276881010, 1.94846223089434035303181385596, 2.88968093158839209239332332866, 3.36876173349330510508749482632, 4.42227396723566730212383477374, 5.12293531194653313703427967852, 5.54885712581227874221666335973, 6.33729312990250068060410043417, 7.31859494765934827293951849257, 7.73782636687158408558992080757