L(s) = 1 | + (0.866 + 1.5i)11-s − 1.73i·17-s + i·19-s + (0.5 + 0.866i)25-s + (1.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.5 − 0.866i)49-s + (0.866 − 1.5i)59-s + (0.866 + 0.5i)67-s − 73-s + (−0.5 − 0.866i)97-s − 1.73·107-s + ⋯ |
L(s) = 1 | + (0.866 + 1.5i)11-s − 1.73i·17-s + i·19-s + (0.5 + 0.866i)25-s + (1.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.5 − 0.866i)49-s + (0.866 − 1.5i)59-s + (0.866 + 0.5i)67-s − 73-s + (−0.5 − 0.866i)97-s − 1.73·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.184586110\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.184586110\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + 1.73iT - T^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.638747944454290640573671773248, −8.945877022897747987736830421919, −7.86234058219858489777945315415, −7.15245277434040327652695126551, −6.55972020214753527237955491411, −5.38548376986792866028673814145, −4.65018812240791955363432122956, −3.74238518414248299572465887110, −2.57931280896920557197045451268, −1.41865516638074689798275351435,
1.08263230468492423744135954629, 2.52140890925723676166638156250, 3.62340435150570997259839553781, 4.32185183457623962873102540737, 5.58804171115562209539678773613, 6.21040649592273343574282564953, 6.94726852136030717653486108241, 8.077526195479491327864461713745, 8.694706856154811500981017601888, 9.234730596204900980851926965191