L(s) = 1 | + (−0.866 + 1.11i)2-s + (−1.58 + 0.707i)3-s + (−0.500 − 1.93i)4-s + 2.44i·5-s + (0.578 − 2.38i)6-s + (2.59 + 1.11i)8-s + (2.00 − 2.23i)9-s + (−2.73 − 2.12i)10-s + 3.46·11-s + (2.15 + 2.70i)12-s + 5.47·13-s + (−1.73 − 3.87i)15-s + (−3.5 + 1.93i)16-s + 4.89i·17-s + (0.767 + 4.17i)18-s − 4.24i·19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.790i)2-s + (−0.912 + 0.408i)3-s + (−0.250 − 0.968i)4-s + 1.09i·5-s + (0.236 − 0.971i)6-s + (0.918 + 0.395i)8-s + (0.666 − 0.745i)9-s + (−0.866 − 0.670i)10-s + 1.04·11-s + (0.623 + 0.781i)12-s + 1.51·13-s + (−0.447 − 0.999i)15-s + (−0.875 + 0.484i)16-s + 1.18i·17-s + (0.181 + 0.983i)18-s − 0.973i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.363854 + 0.755567i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.363854 + 0.755567i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 1.11i)T \) |
| 3 | \( 1 + (1.58 - 0.707i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 7.74iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 + 5.47T + 61T^{2} \) |
| 67 | \( 1 + 7.74iT - 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 15.4iT - 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.97186089660600431943942948573, −10.21975090750974765068526747148, −9.257454245265769166781952060800, −8.440820041400954308701049439361, −7.14328299932412820133835637464, −6.37652027195887687829112197561, −6.04797357748471929800677221002, −4.63567885465901024786134450878, −3.54791071749851257611475941054, −1.33258829372960725565371274200,
0.801196028497204132373535914887, 1.71084682916194469209399284819, 3.67357088040230140214511148401, 4.63549487106539622557994473164, 5.77212866239104754682356852745, 6.85832760334936940106056344445, 7.920386583232763437780254286306, 8.799121936671760350321131230302, 9.474330835745778827626918171777, 10.53051947261554251405098283571