L(s) = 1 | + (1.09 + 1.88i)2-s + (−0.551 + 1.64i)3-s + (−1.37 + 2.38i)4-s + (−0.5 + 0.866i)5-s + (−3.70 + 0.748i)6-s + (0.5 + 0.866i)7-s − 1.64·8-s + (−2.39 − 1.81i)9-s − 2.18·10-s + (1.12 + 1.94i)11-s + (−3.15 − 3.57i)12-s + (2.15 − 3.73i)13-s + (−1.09 + 1.88i)14-s + (−1.14 − 1.29i)15-s + (0.959 + 1.66i)16-s − 7.61·17-s + ⋯ |
L(s) = 1 | + (0.770 + 1.33i)2-s + (−0.318 + 0.947i)3-s + (−0.688 + 1.19i)4-s + (−0.223 + 0.387i)5-s + (−1.51 + 0.305i)6-s + (0.188 + 0.327i)7-s − 0.582·8-s + (−0.797 − 0.603i)9-s − 0.689·10-s + (0.337 + 0.585i)11-s + (−0.911 − 1.03i)12-s + (0.597 − 1.03i)13-s + (−0.291 + 0.504i)14-s + (−0.295 − 0.335i)15-s + (0.239 + 0.415i)16-s − 1.84·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0410517 - 1.64818i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0410517 - 1.64818i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.551 - 1.64i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.09 - 1.88i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.12 - 1.94i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.15 + 3.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 7.61T + 17T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 23 | \( 1 + (-3.45 + 5.98i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.17 - 8.96i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.920 - 1.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.36T + 37T^{2} \) |
| 41 | \( 1 + (3.46 - 5.99i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.98 + 5.16i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.59 - 2.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 4.14T + 53T^{2} \) |
| 59 | \( 1 + (1.41 - 2.45i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.59 + 9.69i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.966 + 1.67i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.17T + 71T^{2} \) |
| 73 | \( 1 - 3.51T + 73T^{2} \) |
| 79 | \( 1 + (3.74 + 6.48i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.16 + 10.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + (-0.988 - 1.71i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.30078585125823486239459966083, −11.07701425191537870250534838776, −10.45014147795024695951245494925, −9.018689742729941195147407728281, −8.287549631147247264872518075082, −6.92862583438591568622958794288, −6.26529074241672748321967583735, −5.08982155167321907617574831474, −4.45139849130052264924278814369, −3.16060463734822689001068104319,
1.08642959768917448214648727422, 2.32535085296588022991597840837, 3.82795942939492103326111754911, 4.81788060278924122248555450825, 6.08683881242967049391560131494, 7.19410909856888201351180151786, 8.445726511299452779308884295706, 9.468381881084152683517154556827, 10.90482918108145505102215874964, 11.48990557004705931271196270829