L(s) = 1 | + (1.27 + 1.16i)3-s + (−2.07 − 0.840i)5-s + (3.28 + 1.89i)7-s + (0.264 + 2.98i)9-s + (−1.86 + 3.23i)11-s + (1.09 − 0.634i)13-s + (−1.66 − 3.49i)15-s + 0.973i·17-s + 2.74·19-s + (1.97 + 6.26i)21-s + (5.83 − 3.36i)23-s + (3.58 + 3.48i)25-s + (−3.15 + 4.12i)27-s + (−1.99 + 3.45i)29-s + (−5.36 − 9.28i)31-s + ⋯ |
L(s) = 1 | + (0.737 + 0.675i)3-s + (−0.926 − 0.375i)5-s + (1.24 + 0.716i)7-s + (0.0882 + 0.996i)9-s + (−0.562 + 0.974i)11-s + (0.304 − 0.176i)13-s + (−0.429 − 0.902i)15-s + 0.236i·17-s + 0.629·19-s + (0.431 + 1.36i)21-s + (1.21 − 0.702i)23-s + (0.717 + 0.696i)25-s + (−0.607 + 0.794i)27-s + (−0.370 + 0.641i)29-s + (−0.962 − 1.66i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36673 + 0.837225i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36673 + 0.837225i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.27 - 1.16i)T \) |
| 5 | \( 1 + (2.07 + 0.840i)T \) |
good | 7 | \( 1 + (-3.28 - 1.89i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.86 - 3.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.09 + 0.634i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 0.973iT - 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 + (-5.83 + 3.36i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.99 - 3.45i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.36 + 9.28i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.59iT - 37T^{2} \) |
| 41 | \( 1 + (2.92 + 5.07i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.95 + 2.86i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.18 - 2.41i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 + (3.98 + 6.91i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.29 + 7.44i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.993 - 0.573i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.20T + 71T^{2} \) |
| 73 | \( 1 + 0.990iT - 73T^{2} \) |
| 79 | \( 1 + (-7.83 + 13.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.08 + 0.625i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 8.97T + 89T^{2} \) |
| 97 | \( 1 + (-5.56 - 3.21i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.43830020985548921713527003587, −10.81155015291015364333806697021, −9.617855893628769107552312897231, −8.673737993124648577275681829237, −8.068299674011016011555457291455, −7.26437688355232340619386580934, −5.22033927525894805198030519183, −4.73278716803567409097340877267, −3.47777130256337791108794421852, −2.02211543497790164527036727395,
1.17697316473679204308597369194, 2.97318882738867386672803053205, 3.95624566877249393364321645991, 5.33490689304147271098150980318, 6.93098786737945455679802997985, 7.58877901125249527775362649665, 8.258291011192870636511464967254, 9.127552390405448931069295233804, 10.70182619865454276409882132695, 11.23933572745250448236659887966