L(s) = 1 | + (−0.5 − 0.866i)5-s + (−1.5 + 2.59i)7-s + (1 − 1.73i)11-s + (1 + 1.73i)13-s − 4·17-s + 8·19-s + (−1.5 − 2.59i)23-s + (−0.499 + 0.866i)25-s + (−0.5 + 0.866i)29-s + 3·35-s − 4·37-s + (2.5 + 4.33i)41-s + (−4 + 6.92i)43-s + (−3.5 + 6.06i)47-s + (−1 − 1.73i)49-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (−0.566 + 0.981i)7-s + (0.301 − 0.522i)11-s + (0.277 + 0.480i)13-s − 0.970·17-s + 1.83·19-s + (−0.312 − 0.541i)23-s + (−0.0999 + 0.173i)25-s + (−0.0928 + 0.160i)29-s + 0.507·35-s − 0.657·37-s + (0.390 + 0.676i)41-s + (−0.609 + 1.05i)43-s + (−0.510 + 0.884i)47-s + (−0.142 − 0.247i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.244832003\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.244832003\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.182333552587743522879109340350, −8.637045339356351693429894147685, −7.79714666363497556063716786405, −6.80146954073009393084333723606, −6.09972126679515788254286482328, −5.33859822947697229457739418892, −4.41520278927015778674928862294, −3.39753157169236030375758788848, −2.53992094950022295776080539180, −1.18913361725541306557264354993,
0.48109018104607831584581149190, 1.90893111527218360871266075467, 3.30214926400492194579736855240, 3.76493951923230503918851462601, 4.85036539563301627219381364584, 5.76885403874693467414759487386, 6.86076558467772617829355139205, 7.15963232031711822304763004830, 7.999028363172932456784622756375, 8.972851750246153059371661478709