L(s) = 1 | + (−0.720 + 1.21i)2-s + (0.399 − 1.68i)3-s + (−0.962 − 1.75i)4-s + (2.09 − 2.09i)5-s + (1.76 + 1.70i)6-s + 7-s + (2.82 + 0.0901i)8-s + (−2.68 − 1.34i)9-s + (1.04 + 4.06i)10-s + (−3.61 − 3.61i)11-s + (−3.33 + 0.922i)12-s + (−2.99 + 2.99i)13-s + (−0.720 + 1.21i)14-s + (−2.69 − 4.37i)15-s + (−2.14 + 3.37i)16-s − 2.25i·17-s + ⋯ |
L(s) = 1 | + (−0.509 + 0.860i)2-s + (0.230 − 0.973i)3-s + (−0.481 − 0.876i)4-s + (0.938 − 0.938i)5-s + (0.719 + 0.694i)6-s + 0.377·7-s + (0.999 + 0.0318i)8-s + (−0.893 − 0.449i)9-s + (0.329 + 1.28i)10-s + (−1.08 − 1.08i)11-s + (−0.963 + 0.266i)12-s + (−0.830 + 0.830i)13-s + (−0.192 + 0.325i)14-s + (−0.696 − 1.12i)15-s + (−0.536 + 0.844i)16-s − 0.545i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.912619 - 0.618784i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.912619 - 0.618784i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.720 - 1.21i)T \) |
| 3 | \( 1 + (-0.399 + 1.68i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-2.09 + 2.09i)T - 5iT^{2} \) |
| 11 | \( 1 + (3.61 + 3.61i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.99 - 2.99i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.25iT - 17T^{2} \) |
| 19 | \( 1 + (-2.89 - 2.89i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.47iT - 23T^{2} \) |
| 29 | \( 1 + (-7.29 - 7.29i)T + 29iT^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 + (-2.12 - 2.12i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.93T + 41T^{2} \) |
| 43 | \( 1 + (0.598 - 0.598i)T - 43iT^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 + (-2.98 + 2.98i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.85 + 1.85i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.40 + 3.40i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.71 - 7.71i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.99iT - 71T^{2} \) |
| 73 | \( 1 - 9.70iT - 73T^{2} \) |
| 79 | \( 1 + 5.37iT - 79T^{2} \) |
| 83 | \( 1 + (4.46 - 4.46i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.75T + 89T^{2} \) |
| 97 | \( 1 - 3.98T + 97T^{2} \) |
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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
−11.37384841354701607064252597917, −10.14513182002464896773778915021, −9.185869379619972139538958327851, −8.452626037432872433688774985494, −7.71804819960997077244156921802, −6.58504837382051024491065058815, −5.63648092137300486947191716253, −4.86750014750949878698029487335, −2.38375803804585258383017197345, −0.919947591685598365243776040068,
2.30745423162935060489727026609, 3.01310925729361420563189579818, 4.57881273081239950933803486638, 5.51739175211050211073178519721, 7.29586168099438692026801458873, 8.116143822978889113117892042187, 9.439142615247450070339742322333, 10.05474620576839691641793452373, 10.46609304277391639943190871553, 11.36142646728890651493464454310