L(s) = 1 | + (1.36 − 0.366i)2-s + (−1.36 − 0.366i)3-s + (1.73 − i)4-s + (−1.36 + 0.366i)5-s − 2·6-s + (1.99 − 2i)8-s + (−0.866 − 0.5i)9-s + (−1.73 + i)10-s + (0.366 − 1.36i)11-s + (−2.73 + 0.732i)12-s + (1 − i)13-s + 2·15-s + (1.99 − 3.46i)16-s + (−1 − 1.73i)17-s + (−1.36 − 0.366i)18-s + (−1.09 − 4.09i)19-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.788 − 0.211i)3-s + (0.866 − 0.5i)4-s + (−0.610 + 0.163i)5-s − 0.816·6-s + (0.707 − 0.707i)8-s + (−0.288 − 0.166i)9-s + (−0.547 + 0.316i)10-s + (0.110 − 0.411i)11-s + (−0.788 + 0.211i)12-s + (0.277 − 0.277i)13-s + 0.516·15-s + (0.499 − 0.866i)16-s + (−0.242 − 0.420i)17-s + (−0.321 − 0.0862i)18-s + (−0.251 − 0.940i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.637806 - 1.30698i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.637806 - 1.30698i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.366i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.36 + 0.366i)T + (2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (1.36 - 0.366i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.366 + 1.36i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.09 + 4.09i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 3i)T - 29iT^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.09 - 1.09i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-5 - 5i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.83 - 6.83i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.09 + 4.09i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.29 - 12.2i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-6.83 - 1.83i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 + i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.46 - 2i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.44953683997457855633812806059, −9.255716778847940030909001990245, −8.069476125818346261938169996748, −7.09266228288039239077231175608, −6.24045477026544824353755734131, −5.60939877429752001293440665617, −4.51609685287121078803663308180, −3.59611010321157134259589652330, −2.42153094949907804958072285165, −0.56495862658599632052662303324,
1.94805754406740269319251515750, 3.54977982277054317555619917832, 4.30296248885310855361922526969, 5.28190650925256759172563825635, 6.01416840664914436388468621447, 6.89127760919081632955773886059, 7.901618339839388105951051010645, 8.613235494106915330104761898068, 10.09024142914215116793362636902, 10.86526175226527659985543761143