L(s) = 1 | + (0.866 − 1.5i)3-s + (1.73 − i)4-s + (−2.86 − 0.767i)7-s + (−1.5 − 2.59i)9-s − 3.46i·12-s + (1.99 − 3.46i)16-s + (2.09 − 7.83i)19-s + (−3.63 + 3.63i)21-s + 5i·25-s − 5.19·27-s + (−5.73 + 1.53i)28-s + (7.83 + 7.83i)31-s + (−5.19 − 3i)36-s + (0.562 + 2.09i)37-s + (1.5 − 0.866i)43-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)3-s + (0.866 − 0.5i)4-s + (−1.08 − 0.290i)7-s + (−0.5 − 0.866i)9-s − 0.999i·12-s + (0.499 − 0.866i)16-s + (0.481 − 1.79i)19-s + (−0.792 + 0.792i)21-s + i·25-s − 1.00·27-s + (−1.08 + 0.290i)28-s + (1.40 + 1.40i)31-s + (−0.866 − 0.5i)36-s + (0.0924 + 0.344i)37-s + (0.228 − 0.132i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09114 - 1.35760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09114 - 1.35760i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.73 + i)T^{2} \) |
| 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 + (2.86 + 0.767i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.09 + 7.83i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.83 - 7.83i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.562 - 2.09i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 0.866i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-4.33 - 7.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.767 + 0.205i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-9.36 + 9.36i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (4.40 - 16.4i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.70147111213447468699955508537, −9.694960756726374850965612607226, −8.966241053185837111601959065232, −7.70566186509270676498625769933, −6.81655458109783170326671294122, −6.48410613122804632727448893041, −5.18522340265245109767038366569, −3.34127778445613368961374692900, −2.55019411188280531281553922488, −0.991659761116170447302090637591,
2.32081466924551677437863181310, 3.27668625423439849235858893490, 4.13466662667870497067700669186, 5.71963794326318497165705828476, 6.47988386140100813457565262383, 7.78076408619146955936590057979, 8.379302278604446214232087726758, 9.650631622861666911482029533024, 10.07565046565347566256026372700, 11.08356426396881625999583040458