L(s) = 1 | + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)5-s + (0.866 + 1.5i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)15-s − 1.73i·21-s + (0.866 − 1.5i)23-s + (−0.499 − 0.866i)25-s − 0.999i·27-s + (0.5 + 0.866i)29-s + 1.73·35-s + (−0.5 + 0.866i)41-s + 0.999·45-s + (−0.866 − 1.5i)47-s + (−1 + 1.73i)49-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)5-s + (0.866 + 1.5i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)15-s − 1.73i·21-s + (0.866 − 1.5i)23-s + (−0.499 − 0.866i)25-s − 0.999i·27-s + (0.5 + 0.866i)29-s + 1.73·35-s + (−0.5 + 0.866i)41-s + 0.999·45-s + (−0.866 − 1.5i)47-s + (−1 + 1.73i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8736490765\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8736490765\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−10.73187450691698984967987388160, −9.697155589634592988573130281146, −8.608516016266899460232474759795, −8.288812447762607878337883643737, −6.87893465749138443788811077912, −5.98189069661683671444574195067, −5.16351289662131739329509394429, −4.69089718754580284850893298477, −2.51949460337761851653115025744, −1.45548708902427059096773852914,
1.43148663123145049578658092340, 3.32006138099948839028592176182, 4.31183165505352995031265383958, 5.22837144868878712016099572722, 6.26115072109761079650799173810, 7.12899306648116238315987085785, 7.77253046960430871630312415362, 9.290952778329490521865457765717, 10.05950253464592205343062015736, 10.75302865327000930474196888415