L(s) = 1 | − i·3-s + 3.46i·5-s + 1.73·7-s + 2·9-s + 5.19i·13-s + 3.46·15-s − 3·17-s + i·19-s − 1.73i·21-s − 1.73·23-s − 6.99·25-s − 5i·27-s + 1.73i·29-s − 3.46·31-s + 5.99i·35-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.54i·5-s + 0.654·7-s + 0.666·9-s + 1.44i·13-s + 0.894·15-s − 0.727·17-s + 0.229i·19-s − 0.377i·21-s − 0.361·23-s − 1.39·25-s − 0.962i·27-s + 0.321i·29-s − 0.622·31-s + 1.01i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.662261103\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.662261103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 - 1.73iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 5.19iT - 53T^{2} \) |
| 59 | \( 1 + 9iT - 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.995721088353125949789633820881, −9.130809014736659049190645427091, −8.044909922531428311511242723150, −7.22872877430361270799768076739, −6.77831619077831555211479370128, −6.04354071914397119737973510857, −4.62019438664337834838522489439, −3.78146532603560973958805244249, −2.45550167055819547236929695384, −1.66640161108249803549929747609,
0.73052321173957581962420246312, 1.99038635616161493495047140663, 3.63761215572347329031137936426, 4.56267758728510822804505539546, 5.08185145654037014623175533490, 5.90208490320469562409586990325, 7.33411712295664014278373107936, 8.067180306272979469618645743157, 8.838165465216493868353663711785, 9.420027115361187682865606078067