L(s) = 1 | + (−0.281 − 1.04i)2-s + (−0.866 − 0.5i)3-s + (0.710 − 0.410i)4-s + (−1.02 − 1.02i)5-s + (−0.281 + 1.04i)6-s + (2.23 − 1.41i)7-s + (−2.16 − 2.16i)8-s + (0.499 + 0.866i)9-s + (−0.788 + 1.36i)10-s + (0.972 − 0.260i)11-s − 0.820·12-s + (−3.05 + 1.91i)13-s + (−2.11 − 1.94i)14-s + (0.376 + 1.40i)15-s + (−0.842 + 1.45i)16-s + (−2.37 − 4.10i)17-s + ⋯ |
L(s) = 1 | + (−0.198 − 0.741i)2-s + (−0.499 − 0.288i)3-s + (0.355 − 0.205i)4-s + (−0.459 − 0.459i)5-s + (−0.114 + 0.428i)6-s + (0.843 − 0.536i)7-s + (−0.765 − 0.765i)8-s + (0.166 + 0.288i)9-s + (−0.249 + 0.432i)10-s + (0.293 − 0.0785i)11-s − 0.236·12-s + (−0.847 + 0.530i)13-s + (−0.565 − 0.519i)14-s + (0.0970 + 0.362i)15-s + (−0.210 + 0.364i)16-s + (−0.574 − 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.272503 - 0.940819i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.272503 - 0.940819i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
| 13 | \( 1 + (3.05 - 1.91i)T \) |
good | 2 | \( 1 + (0.281 + 1.04i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (1.02 + 1.02i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.972 + 0.260i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (2.37 + 4.10i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.391 + 1.46i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (0.337 + 0.194i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.30 - 7.44i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.92 - 2.92i)T + 31iT^{2} \) |
| 37 | \( 1 + (-9.77 + 2.61i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-8.64 + 2.31i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (8.28 - 4.78i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.78 + 4.78i)T - 47iT^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + (-0.889 - 0.238i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-8.36 + 4.82i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.71 - 6.39i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (2.56 + 0.687i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (2.46 - 2.46i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.06T + 79T^{2} \) |
| 83 | \( 1 + (-2.43 - 2.43i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.79 - 10.4i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-3.55 + 13.2i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.52224272658943402834826881497, −10.87652322598148860040638406293, −9.817790023512348930975248645003, −8.796262236041642074485169172470, −7.44972409652166124654536303403, −6.73317692344992429245524861407, −5.22256410882269443509175753918, −4.17608647495530240206943412138, −2.35062658224171728784779631024, −0.861620073255670591240264023178,
2.42549742520588268221457450664, 4.10562371460976574793981795297, 5.48786878227653781066709834446, 6.32129038783945177403563397269, 7.52761964042156651241375107231, 8.098452344670755847539817335674, 9.304134445091425973373516574647, 10.56108282231685683177475567612, 11.52002123852831999714256492463, 11.88871831099115786127621644582