L(s) = 1 | + 3.55i·2-s + (−1.69 + 2.47i)3-s − 8.66·4-s − 6.82i·5-s + (−8.81 − 6.02i)6-s + 0.148·7-s − 16.5i·8-s + (−3.26 − 8.38i)9-s + 24.2·10-s + 1.80i·11-s + (14.6 − 21.4i)12-s − 17.1·13-s + 0.528i·14-s + (16.8 + 11.5i)15-s + 24.3·16-s − 18.3i·17-s + ⋯ |
L(s) = 1 | + 1.77i·2-s + (−0.564 + 0.825i)3-s − 2.16·4-s − 1.36i·5-s + (−1.46 − 1.00i)6-s + 0.0212·7-s − 2.07i·8-s + (−0.363 − 0.931i)9-s + 2.42·10-s + 0.164i·11-s + (1.22 − 1.78i)12-s − 1.31·13-s + 0.0377i·14-s + (1.12 + 0.770i)15-s + 1.52·16-s − 1.07i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.156074 - 0.0482438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.156074 - 0.0482438i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.69 - 2.47i)T \) |
| 59 | \( 1 - 7.68iT \) |
good | 2 | \( 1 - 3.55iT - 4T^{2} \) |
| 5 | \( 1 + 6.82iT - 25T^{2} \) |
| 7 | \( 1 - 0.148T + 49T^{2} \) |
| 11 | \( 1 - 1.80iT - 121T^{2} \) |
| 13 | \( 1 + 17.1T + 169T^{2} \) |
| 17 | \( 1 + 18.3iT - 289T^{2} \) |
| 19 | \( 1 + 19.6T + 361T^{2} \) |
| 23 | \( 1 - 40.3iT - 529T^{2} \) |
| 29 | \( 1 + 33.8iT - 841T^{2} \) |
| 31 | \( 1 + 46.0T + 961T^{2} \) |
| 37 | \( 1 - 40.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 21.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 4.07T + 1.84e3T^{2} \) |
| 47 | \( 1 + 45.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 29.4iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 28.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.55T + 4.48e3T^{2} \) |
| 71 | \( 1 + 11.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 113.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 32.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 145. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 148. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 133.T + 9.40e3T^{2} \) |
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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
−12.60072765243956357571252404723, −11.51274542474059765268444625099, −9.640559105339892618506934230429, −9.350459981208556272812760922671, −8.170030417150509097648334872126, −7.10204827276295173647934503674, −5.71959656671048741346569198210, −5.03209314029665607402520409265, −4.25334742704471340699668234975, −0.10162709168686636136030408094,
1.98646063558912228310615912089, 2.94155994417372611549768797491, 4.54274754168535961174701538815, 6.18965433917956431352376145466, 7.36348249567530651114886486675, 8.740195255925333141688331870062, 10.26728997916418017856939954773, 10.70366112213110789498307417889, 11.42395336607779734250773400175, 12.62903163786653058334117844900