L(s) = 1 | + (−0.806 − 1.16i)2-s + i·3-s + (−0.699 + 1.87i)4-s + (1.16 − 0.806i)6-s − 0.746·7-s + (2.74 − 0.699i)8-s − 9-s + 5.36i·11-s + (−1.87 − 0.699i)12-s − 2.92i·13-s + (0.601 + 0.866i)14-s + (−3.02 − 2.62i)16-s − 2.13·17-s + (0.806 + 1.16i)18-s + 1.73i·19-s + ⋯ |
L(s) = 1 | + (−0.570 − 0.821i)2-s + 0.577i·3-s + (−0.349 + 0.936i)4-s + (0.474 − 0.329i)6-s − 0.282·7-s + (0.968 − 0.247i)8-s − 0.333·9-s + 1.61i·11-s + (−0.540 − 0.201i)12-s − 0.811i·13-s + (0.160 + 0.231i)14-s + (−0.755 − 0.655i)16-s − 0.517·17-s + (0.190 + 0.273i)18-s + 0.397i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.342859 + 0.441297i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.342859 + 0.441297i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.806 + 1.16i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.746T + 7T^{2} \) |
| 11 | \( 1 - 5.36iT - 11T^{2} \) |
| 13 | \( 1 + 2.92iT - 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 - 6.74iT - 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 1.07iT - 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 7.44iT - 43T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 - 7.72iT - 53T^{2} \) |
| 59 | \( 1 - 6.85iT - 59T^{2} \) |
| 61 | \( 1 + 6.45iT - 61T^{2} \) |
| 67 | \( 1 - 7.44iT - 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.690T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 + 5.85iT - 83T^{2} \) |
| 89 | \( 1 - 7.59T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−10.62675341297917817811957433306, −10.04863750691141834805443638203, −9.514631783576072444313604290402, −8.448739400445218208854029071417, −7.66007679753556324749607342123, −6.56886070397162498410462174341, −5.05602147133902731873060567504, −4.18085000834518565530146837729, −3.09348413190969811732507335812, −1.83098214645737332572255802108,
0.36412827931720773684822848800, 2.05918041610614081917854046339, 3.77950276141966691808917156716, 5.17890563335920392732985750429, 6.27914015174090946836960195394, 6.61940670336845572818184136441, 7.915092357693926310956011629056, 8.457819602683363182750781745326, 9.316793443123483233770848648248, 10.24913880386126223497420284432