L(s) = 1 | + (−0.382 − 0.923i)2-s + (−1.27 + 0.253i)3-s + (−0.707 + 0.707i)4-s + (−1.36 − 1.77i)5-s + (0.722 + 1.08i)6-s + (−0.128 − 0.192i)7-s + (0.923 + 0.382i)8-s + (−1.21 + 0.501i)9-s + (−1.11 + 1.93i)10-s + (−3.12 + 2.08i)11-s + (0.722 − 1.08i)12-s − 6.31·13-s + (−0.128 + 0.192i)14-s + (2.18 + 1.91i)15-s − i·16-s + (−2.15 − 3.51i)17-s + ⋯ |
L(s) = 1 | + (−0.270 − 0.653i)2-s + (−0.735 + 0.146i)3-s + (−0.353 + 0.353i)4-s + (−0.609 − 0.792i)5-s + (0.294 + 0.441i)6-s + (−0.0486 − 0.0728i)7-s + (0.326 + 0.135i)8-s + (−0.403 + 0.167i)9-s + (−0.352 + 0.612i)10-s + (−0.941 + 0.629i)11-s + (0.208 − 0.311i)12-s − 1.75·13-s + (−0.0344 + 0.0514i)14-s + (0.564 + 0.494i)15-s − 0.250i·16-s + (−0.522 − 0.852i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0270941 + 0.147959i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0270941 + 0.147959i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 + 0.923i)T \) |
| 5 | \( 1 + (1.36 + 1.77i)T \) |
| 17 | \( 1 + (2.15 + 3.51i)T \) |
good | 3 | \( 1 + (1.27 - 0.253i)T + (2.77 - 1.14i)T^{2} \) |
| 7 | \( 1 + (0.128 + 0.192i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (3.12 - 2.08i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + 6.31T + 13T^{2} \) |
| 19 | \( 1 + (-5.28 - 2.18i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.67 + 8.43i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.221 - 1.11i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (1.28 + 0.856i)T + (11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (-0.0492 - 0.247i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.06 + 5.36i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (3.68 - 8.89i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 4.58iT - 47T^{2} \) |
| 53 | \( 1 + (5.77 - 2.39i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (0.547 + 1.32i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (10.1 + 2.02i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (0.973 + 0.973i)T + 67iT^{2} \) |
| 71 | \( 1 + (1.06 - 1.59i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (1.07 - 1.60i)T + (-27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.466 - 0.698i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-5.53 - 13.3i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (9.96 - 9.96i)T - 89iT^{2} \) |
| 97 | \( 1 + (-2.79 + 4.18i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−12.19069371399064500452754588363, −11.34087443350351077871689890683, −10.29844561895748051520254859437, −9.397864399611410904712821926263, −8.146380735905079619423548701818, −7.18734347589421207467362734084, −5.18721042707378425162113211566, −4.67066966545173751640802662183, −2.66288771691891477173600793561, −0.15728281895869833293073069333,
3.05876507751151419401197867131, 4.98401394895833610788359854230, 5.95402320943128959401553594023, 7.17037616051431025835998291573, 7.84113262829201099901949809303, 9.265312783568351146223411943816, 10.43451357694445455169809746179, 11.31822771318031225894253184562, 12.13348642452547535749554405556, 13.41156010374559298207249714792