L(s) = 1 | + 1.11·5-s + (−1.37 + 2.26i)7-s − 0.812·11-s + 2·13-s + 3.55i·17-s − 4.52i·19-s − 3.63i·23-s − 3.75·25-s + 8.95i·29-s − 5.08·31-s + (−1.53 + 2.52i)35-s + 0.718i·37-s + 10.4i·41-s + 1.11·43-s + 8.55·47-s + ⋯ |
L(s) = 1 | + 0.498·5-s + (−0.518 + 0.854i)7-s − 0.245·11-s + 0.554·13-s + 0.862i·17-s − 1.03i·19-s − 0.758i·23-s − 0.751·25-s + 1.66i·29-s − 0.914·31-s + (−0.258 + 0.426i)35-s + 0.118i·37-s + 1.63i·41-s + 0.170·43-s + 1.24·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.056216052\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.056216052\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.37 - 2.26i)T \) |
good | 5 | \( 1 - 1.11T + 5T^{2} \) |
| 11 | \( 1 + 0.812T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.55iT - 17T^{2} \) |
| 19 | \( 1 + 4.52iT - 19T^{2} \) |
| 23 | \( 1 + 3.63iT - 23T^{2} \) |
| 29 | \( 1 - 8.95iT - 29T^{2} \) |
| 31 | \( 1 + 5.08T + 31T^{2} \) |
| 37 | \( 1 - 0.718iT - 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 - 1.11T + 43T^{2} \) |
| 47 | \( 1 - 8.55T + 47T^{2} \) |
| 53 | \( 1 - 5.08iT - 53T^{2} \) |
| 59 | \( 1 - 2.23iT - 59T^{2} \) |
| 61 | \( 1 - 5.27T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 5.87iT - 71T^{2} \) |
| 73 | \( 1 - 3.86iT - 73T^{2} \) |
| 79 | \( 1 - 1.70iT - 79T^{2} \) |
| 83 | \( 1 - 7.27iT - 83T^{2} \) |
| 89 | \( 1 + 3.37iT - 89T^{2} \) |
| 97 | \( 1 + 8.55iT - 97T^{2} \) |
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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
−8.877780034240087137927480830058, −8.114326530975171893694304586662, −7.14747554667080398991847376894, −6.39174352521593641358052342842, −5.82141408435978332016306628593, −5.13069543245349241813532731035, −4.14447033083242216963169350957, −3.14687698555668675139439737873, −2.41236765596003841162529901204, −1.37576482807414501065882195864,
0.29376694343137726575926845960, 1.56042300046404868427581920327, 2.57281981010476294184996595781, 3.69801660414401146521958575335, 4.12121611692954242805494930970, 5.41613643506916836045933398709, 5.85744783767649351633033221142, 6.72026272445872391627327363303, 7.50142518298431981738469837582, 7.978612439536502563548658395715