L(s) = 1 | + (−0.5 + 0.866i)7-s + 2·13-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)31-s + (−1 − 1.73i)37-s − 43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (0.5 − 0.866i)73-s + (−1 − 1.73i)79-s + (−1 + 1.73i)91-s − 97-s + (−1 − 1.73i)103-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)7-s + 2·13-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)31-s + (−1 − 1.73i)37-s − 43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (0.5 − 0.866i)73-s + (−1 − 1.73i)79-s + (−1 + 1.73i)91-s − 97-s + (−1 − 1.73i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9725035391\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9725035391\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - 2T + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.62932871528426230517026056625, −9.687532573179586339968001977749, −8.870239416831465612952893521537, −8.232794775295628069526212849968, −7.10171719857339533757056547881, −5.96747182206304400887987792285, −5.61043948651723752723535650071, −4.00545553845051295599186923696, −3.18449586044372927020330116682, −1.68767077795690034784893235093,
1.26918191813805124151530000226, 3.10468264816957414902063507855, 3.92259897083775955625535401179, 5.04877944921169936566024468690, 6.34137702286563838020390891255, 6.78871695268480433648327619841, 8.034532254963809259935478754537, 8.699164035819916900282065218251, 9.742232412987617180579576569954, 10.50220222348165524604662537221