L(s) = 1 | + (−0.0749 + 0.0200i)3-s + (−0.965 + 2.01i)5-s + (1.45 + 2.20i)7-s + (−2.59 + 1.49i)9-s + (−0.390 + 0.677i)11-s + (−3.28 − 3.28i)13-s + (0.0318 − 0.170i)15-s + (1.40 + 5.24i)17-s + (2.91 + 5.05i)19-s + (−0.153 − 0.136i)21-s + (8.39 + 2.24i)23-s + (−3.13 − 3.89i)25-s + (0.328 − 0.328i)27-s − 0.303i·29-s + (−5.04 − 2.91i)31-s + ⋯ |
L(s) = 1 | + (−0.0432 + 0.0115i)3-s + (−0.431 + 0.902i)5-s + (0.551 + 0.834i)7-s + (−0.864 + 0.498i)9-s + (−0.117 + 0.204i)11-s + (−0.911 − 0.911i)13-s + (0.00821 − 0.0440i)15-s + (0.341 + 1.27i)17-s + (0.669 + 1.15i)19-s + (−0.0335 − 0.0297i)21-s + (1.74 + 0.468i)23-s + (−0.627 − 0.778i)25-s + (0.0632 − 0.0632i)27-s − 0.0564i·29-s + (−0.905 − 0.522i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0327 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0327 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.723050 + 0.747137i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.723050 + 0.747137i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.965 - 2.01i)T \) |
| 7 | \( 1 + (-1.45 - 2.20i)T \) |
good | 3 | \( 1 + (0.0749 - 0.0200i)T + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (0.390 - 0.677i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.28 + 3.28i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.40 - 5.24i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.91 - 5.05i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-8.39 - 2.24i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 0.303iT - 29T^{2} \) |
| 31 | \( 1 + (5.04 + 2.91i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.90 + 7.12i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 4.39iT - 41T^{2} \) |
| 43 | \( 1 + (-7.33 + 7.33i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.28 - 0.611i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.61 + 6.01i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.56 + 4.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.91 - 5.14i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.47 - 0.395i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 5.83T + 71T^{2} \) |
| 73 | \( 1 + (-12.9 + 3.47i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-8.66 + 5.00i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.94 - 2.94i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.43 - 11.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0900 - 0.0900i)T - 97iT^{2} \) |
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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
−12.03872309892167943100121969830, −11.08076366151891606694438674119, −10.48021097212039843191446351433, −9.229018141450387582191357223824, −8.009491466307075728609976723575, −7.51786531136834404338917649942, −5.93244025162747008428653681273, −5.20259129510699119931968030093, −3.47658239030681718008113620780, −2.32743365224918834219475010208,
0.816685596772616207771010408214, 3.00684290827041592110931095750, 4.58217209792643988665786924109, 5.19919319704883261117946573713, 6.91461661138328310960773554369, 7.65165782470552871228090207815, 8.947173160640766918926667391537, 9.399375004380451222495334236439, 10.98850387698960394408166766746, 11.57251518332626550905094050238