L(s) = 1 | + (−0.0341 − 0.0592i)2-s + (1.15 + 1.29i)3-s + (0.997 − 1.72i)4-s + 2.66·5-s + (0.0368 − 0.112i)6-s − 0.273·8-s + (−0.329 + 2.98i)9-s + (−0.0910 − 0.157i)10-s − 1.59·11-s + (3.38 − 0.709i)12-s + (2.62 + 4.54i)13-s + (3.07 + 3.43i)15-s + (−1.98 − 3.43i)16-s + (−3.27 − 5.67i)17-s + (0.187 − 0.0824i)18-s + (0.950 − 1.64i)19-s + ⋯ |
L(s) = 1 | + (−0.0241 − 0.0418i)2-s + (0.667 + 0.744i)3-s + (0.498 − 0.864i)4-s + 1.19·5-s + (0.0150 − 0.0459i)6-s − 0.0965·8-s + (−0.109 + 0.993i)9-s + (−0.0287 − 0.0498i)10-s − 0.482·11-s + (0.976 − 0.204i)12-s + (0.728 + 1.26i)13-s + (0.794 + 0.887i)15-s + (−0.496 − 0.859i)16-s + (−0.793 − 1.37i)17-s + (0.0442 − 0.0194i)18-s + (0.218 − 0.377i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.16467 + 0.180600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16467 + 0.180600i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.15 - 1.29i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.0341 + 0.0592i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 2.66T + 5T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 + (-2.62 - 4.54i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.27 + 5.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.950 + 1.64i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 + (3.19 - 5.53i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.35 + 5.81i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.11 - 3.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.69 - 6.40i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.63 + 9.75i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.89 - 3.29i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.44 + 7.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.44 - 9.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.35 - 2.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.66 + 2.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + (-1.09 - 1.90i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.406 + 0.704i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.41 + 5.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.235 + 0.407i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.57 - 4.46i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.96150831022923211971938262368, −10.10480055468745031191596488636, −9.399910967251486647217309827301, −8.909867898154392475993499630422, −7.37004665521108149539408710651, −6.33082221979368955548520956544, −5.40210492497143515545257155775, −4.44373015861423033397638375316, −2.72699202240785047279347868102, −1.84245573626473724460177928099,
1.76889882020280695006672838607, 2.73765343318381396227981359956, 3.86008392131490246677613197479, 5.85596397613546072436328288730, 6.33258487076560844210446527551, 7.62856552304374652245704117323, 8.216202103682097658530200104441, 9.067191765258372254083672893439, 10.22488814214812533908696672813, 11.03410547439670621617109146774