L(s) = 1 | + (0.631 − 2.35i)2-s + (−0.102 − 1.72i)3-s + (−3.42 − 1.97i)4-s + (−4.13 − 0.849i)6-s + (−1.82 − 1.91i)7-s + (−3.36 + 3.36i)8-s + (−2.97 + 0.355i)9-s + (3.08 + 1.77i)11-s + (−3.06 + 6.11i)12-s + (−1.28 − 1.28i)13-s + (−5.67 + 3.08i)14-s + (1.85 + 3.21i)16-s + (2.95 − 0.792i)17-s + (−1.04 + 7.24i)18-s + (−0.331 + 0.191i)19-s + ⋯ |
L(s) = 1 | + (0.446 − 1.66i)2-s + (−0.0593 − 0.998i)3-s + (−1.71 − 0.987i)4-s + (−1.68 − 0.346i)6-s + (−0.688 − 0.725i)7-s + (−1.19 + 1.19i)8-s + (−0.992 + 0.118i)9-s + (0.928 + 0.536i)11-s + (−0.884 + 1.76i)12-s + (−0.356 − 0.356i)13-s + (−1.51 + 0.823i)14-s + (0.463 + 0.803i)16-s + (0.717 − 0.192i)17-s + (−0.245 + 1.70i)18-s + (−0.0761 + 0.0439i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.755013 + 0.983924i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.755013 + 0.983924i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.102 + 1.72i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.82 + 1.91i)T \) |
good | 2 | \( 1 + (-0.631 + 2.35i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (-3.08 - 1.77i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.28 + 1.28i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.95 + 0.792i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.331 - 0.191i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.45 + 0.658i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + (-0.323 + 0.561i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.00 + 1.34i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (-0.335 - 0.335i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.751 + 2.80i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.815 + 3.04i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.81 + 6.60i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.45 + 9.45i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.31 - 12.3i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 3.06iT - 71T^{2} \) |
| 73 | \( 1 + (3.17 - 0.849i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.21 + 1.85i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.973 + 0.973i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.51 - 2.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 10.3i)T - 97iT^{2} \) |
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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
−10.35262761821637142050356900329, −9.750006819712449067674831047005, −8.717827679005904811214849613412, −7.43191965993957365903550667276, −6.54845002512555634568123834089, −5.29657054513758124230290501083, −4.02995419522975743640590787527, −3.10354822190664980883144614782, −1.90143030800085789159101403956, −0.65031121872674125120948607816,
3.15515347363174664539495432337, 4.16743610843460826255688235224, 5.13967943593543390031519612638, 6.05207918963852231264726889232, 6.55655808268987264809832306368, 7.901684769538561853354712007103, 8.788669178420061075667498739432, 9.344221992009166332431822541081, 10.29957935096375186643700999167, 11.67417367260412273513838235229