L(s) = 1 | + 2.50·2-s + (1.71 − 0.207i)3-s + 4.29·4-s + i·5-s + (4.31 − 0.520i)6-s − i·7-s + 5.75·8-s + (2.91 − 0.713i)9-s + 2.50i·10-s + (−3.26 − 0.595i)11-s + (7.38 − 0.890i)12-s − 2.68i·13-s − 2.50i·14-s + (0.207 + 1.71i)15-s + 5.84·16-s − 1.44·17-s + ⋯ |
L(s) = 1 | + 1.77·2-s + (0.992 − 0.119i)3-s + 2.14·4-s + 0.447i·5-s + (1.76 − 0.212i)6-s − 0.377i·7-s + 2.03·8-s + (0.971 − 0.237i)9-s + 0.793i·10-s + (−0.983 − 0.179i)11-s + (2.13 − 0.257i)12-s − 0.744i·13-s − 0.670i·14-s + (0.0535 + 0.443i)15-s + 1.46·16-s − 0.351·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.116246512\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.116246512\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.71 + 0.207i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (3.26 + 0.595i)T \) |
good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 13 | \( 1 + 2.68iT - 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 - 7.09iT - 19T^{2} \) |
| 23 | \( 1 - 0.250iT - 23T^{2} \) |
| 29 | \( 1 + 4.88T + 29T^{2} \) |
| 31 | \( 1 + 4.26T + 31T^{2} \) |
| 37 | \( 1 - 7.50T + 37T^{2} \) |
| 41 | \( 1 + 1.94T + 41T^{2} \) |
| 43 | \( 1 + 7.29iT - 43T^{2} \) |
| 47 | \( 1 + 1.14iT - 47T^{2} \) |
| 53 | \( 1 - 2.58iT - 53T^{2} \) |
| 59 | \( 1 - 11.7iT - 59T^{2} \) |
| 61 | \( 1 - 13.2iT - 61T^{2} \) |
| 67 | \( 1 + 5.66T + 67T^{2} \) |
| 71 | \( 1 + 4.04iT - 71T^{2} \) |
| 73 | \( 1 + 0.248iT - 73T^{2} \) |
| 79 | \( 1 + 14.3iT - 79T^{2} \) |
| 83 | \( 1 - 3.49T + 83T^{2} \) |
| 89 | \( 1 + 11.0iT - 89T^{2} \) |
| 97 | \( 1 + 2.11T + 97T^{2} \) |
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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
−10.20851869007616692035513607298, −8.827304857341281974431518185468, −7.58654566565142737766157015849, −7.45911767275704626853253536496, −6.16795719304664775236488912837, −5.50745883550119933310817722787, −4.34538259031742211850613305651, −3.58381529144474164363297392843, −2.87389141232327783050950234879, −1.90002132692161126038810855298,
2.01030765558351209814738924225, 2.69675661646298638358104477020, 3.70807336244832689987661054418, 4.67840375761071163251119889459, 5.10253058888537497761399169691, 6.31614211906818524229766483885, 7.15310643387999131336931231934, 7.966339120283719792829567593043, 9.021775241545991477480114795503, 9.699871270723552373482007850916