L(s) = 1 | + (0.366 + 1.36i)2-s + (−1.5 + 0.866i)3-s + (−1.73 + i)4-s + (−1.73 − 1.73i)6-s + (−2 + 1.73i)7-s + (−2 − 1.99i)8-s + (0.866 − 0.5i)11-s + (1.73 − 3i)12-s − 3.46·13-s + (−3.09 − 2.09i)14-s + (1.99 − 3.46i)16-s + (0.866 + 1.5i)17-s + (2.59 − 4.5i)19-s + (1.50 − 4.33i)21-s + (1 + 0.999i)22-s + (−0.5 + 0.866i)23-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)3-s + (−0.866 + 0.5i)4-s + (−0.707 − 0.707i)6-s + (−0.755 + 0.654i)7-s + (−0.707 − 0.707i)8-s + (0.261 − 0.150i)11-s + (0.499 − 0.866i)12-s − 0.960·13-s + (−0.827 − 0.560i)14-s + (0.499 − 0.866i)16-s + (0.210 + 0.363i)17-s + (0.596 − 1.03i)19-s + (0.327 − 0.944i)21-s + (0.213 + 0.213i)22-s + (−0.104 + 0.180i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 3 | \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + (-0.866 - 1.5i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.59 + 4.5i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (0.866 + 1.5i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.59 - 1.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (7.5 + 4.33i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14iT - 71T^{2} \) |
| 73 | \( 1 + (-4.33 - 7.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.79 + 4.5i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + (13.5 + 7.79i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−10.03590938697895814323944038614, −9.482950740475853154850262529895, −8.578142902864523547717124533244, −7.47457579605821980622061543019, −6.60719548535132736608479386710, −5.69160011151600175782804437617, −5.16839598387022676127077856254, −4.11056799359014177845823238096, −2.82535537886469655401796563754, 0,
1.35875752573773810669244344562, 2.95585148792669214423101273942, 3.98288927177179315572824353925, 5.14450797898659285188970422895, 6.00802259076958612267629136481, 6.91643814560194720328877919890, 7.917279309222902455252546946437, 9.404063633507386959706419921209, 9.775858298332276670377244079219