L(s) = 1 | + (−0.258 − 1.71i)3-s + (0.445 + 1.94i)4-s + (4.43 − 2.55i)7-s + (−2.86 + 0.884i)9-s + (3.22 − 1.26i)12-s + (−0.242 − 3.23i)13-s + (−3.60 + 1.73i)16-s + (−2.48 + 8.04i)19-s + (−5.52 − 6.92i)21-s + (−1.82 + 4.65i)25-s + (2.25 + 4.68i)27-s + (6.95 + 7.49i)28-s + (−0.879 − 2.24i)31-s + (−3 − 5.19i)36-s + (−8.28 − 4.78i)37-s + ⋯ |
L(s) = 1 | + (−0.149 − 0.988i)3-s + (0.222 + 0.974i)4-s + (1.67 − 0.966i)7-s + (−0.955 + 0.294i)9-s + (0.930 − 0.365i)12-s + (−0.0672 − 0.897i)13-s + (−0.900 + 0.433i)16-s + (−0.568 + 1.84i)19-s + (−1.20 − 1.51i)21-s + (−0.365 + 0.930i)25-s + (0.433 + 0.900i)27-s + (1.31 + 1.41i)28-s + (−0.157 − 0.402i)31-s + (−0.5 − 0.866i)36-s + (−1.36 − 0.786i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12486 - 0.282358i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12486 - 0.282358i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 + 1.71i)T \) |
| 43 | \( 1 + (6.55 - 0.112i)T \) |
good | 2 | \( 1 + (-0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (1.82 - 4.65i)T^{2} \) |
| 7 | \( 1 + (-4.43 + 2.55i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.242 + 3.23i)T + (-12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (-6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (2.48 - 8.04i)T + (-15.6 - 10.7i)T^{2} \) |
| 23 | \( 1 + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (0.879 + 2.24i)T + (-22.7 + 21.0i)T^{2} \) |
| 37 | \( 1 + (8.28 + 4.78i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-14.1 - 5.54i)T + (44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (2.15 + 0.665i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (5.30 - 70.8i)T^{2} \) |
| 73 | \( 1 + (-4.23 + 0.317i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-5.55 - 9.62i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-1.46 + 6.43i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−13.16472901170038651847092117106, −12.23357186635285004586737240773, −11.38835289955029908045613927022, −10.52139174246923011176859695119, −8.381615219053947046166839254323, −7.895364589378192173785196457831, −7.06918269198119880268637563077, −5.42651836448152976696382492410, −3.78124946803131712648921463558, −1.80674298657420727160625500956,
2.20070493198065664713271425922, 4.63946925403758013433613844871, 5.22049094805593552428282074325, 6.58400946024421680496348568341, 8.484802465877643606194873175021, 9.203453462020323818963615954049, 10.45146956231294459674742965827, 11.33120480379141349580150631039, 11.83789940128132414002891707349, 13.85577905285381159426981636279