L(s) = 1 | + (−0.0418 − 1.41i)2-s + (1.46 − 0.925i)3-s + (−1.99 + 0.118i)4-s + (0.5 − 0.866i)5-s + (−1.36 − 2.03i)6-s + (0.947 − 0.546i)7-s + (0.250 + 2.81i)8-s + (1.28 − 2.70i)9-s + (−1.24 − 0.670i)10-s + (2.31 − 1.33i)11-s + (−2.81 + 2.02i)12-s + (−2.12 − 1.22i)13-s + (−0.812 − 1.31i)14-s + (−0.0691 − 1.73i)15-s + (3.97 − 0.472i)16-s + 0.124i·17-s + ⋯ |
L(s) = 1 | + (−0.0295 − 0.999i)2-s + (0.845 − 0.534i)3-s + (−0.998 + 0.0591i)4-s + (0.223 − 0.387i)5-s + (−0.558 − 0.829i)6-s + (0.357 − 0.206i)7-s + (0.0886 + 0.996i)8-s + (0.429 − 0.903i)9-s + (−0.393 − 0.212i)10-s + (0.698 − 0.403i)11-s + (−0.812 + 0.583i)12-s + (−0.589 − 0.340i)13-s + (−0.217 − 0.351i)14-s + (−0.0178 − 0.446i)15-s + (0.993 − 0.118i)16-s + 0.0302i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.690397 - 1.49192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.690397 - 1.49192i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0418 + 1.41i)T \) |
| 3 | \( 1 + (-1.46 + 0.925i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-0.947 + 0.546i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.31 + 1.33i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.124iT - 17T^{2} \) |
| 19 | \( 1 + 5.84T + 19T^{2} \) |
| 23 | \( 1 + (1.21 - 2.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.29 - 7.43i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.54 - 1.46i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.42iT - 37T^{2} \) |
| 41 | \( 1 + (-7.57 - 4.37i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.14 + 3.71i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.32 - 5.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.14T + 53T^{2} \) |
| 59 | \( 1 + (2.34 + 1.35i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.69 + 5.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.707 - 1.22i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 5.35T + 73T^{2} \) |
| 79 | \( 1 + (-2.07 + 1.19i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.2 + 6.47i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 13.3iT - 89T^{2} \) |
| 97 | \( 1 + (-7.51 - 13.0i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.13204025877705635463562670729, −10.17194296525083343872657670660, −9.191145561476719529461899068393, −8.566905994130419307645271778139, −7.67253275454088806933949606603, −6.29281624389347669273823956167, −4.79619470388312815335925204913, −3.72239380461315981922178135680, −2.46000968284655251006583159800, −1.20041843256874950817633542973,
2.30716624194053746569898703935, 4.01276048490080552091170789382, 4.73004903639194539986287813380, 6.13818752734857095737663766925, 7.09401862537193652276227140304, 8.114306592914751382915457044658, 8.849612795825716856649145330904, 9.755797423346569847579247588777, 10.41662342134931157876148698541, 11.83566203561537142657160635874