L(s) = 1 | + (1 + 1.73i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s + (−1.49 + 2.59i)25-s + 2·43-s + (0.999 − 1.73i)45-s + (−0.5 − 0.866i)47-s + 1.99·55-s + (−0.5 − 0.866i)61-s + (−0.5 + 0.866i)73-s + (−0.499 + 0.866i)81-s + 83-s + ⋯ |
L(s) = 1 | + (1 + 1.73i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s + (−1.49 + 2.59i)25-s + 2·43-s + (0.999 − 1.73i)45-s + (−0.5 − 0.866i)47-s + 1.99·55-s + (−0.5 − 0.866i)61-s + (−0.5 + 0.866i)73-s + (−0.499 + 0.866i)81-s + 83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.506546851\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.506546851\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 2T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.050718350554545922196544990352, −8.042345156114895884549894988709, −7.17324994670238977183062446602, −6.48444124048869285804360487881, −5.99129352535923124010317509311, −5.49134708805229452648064660560, −3.82736949648929580698496390120, −3.37487137959198760575014124053, −2.55386706215979386477020079443, −1.47901243647132623168245217188,
0.918652864142979329308802143353, 2.00689714113842994125695937303, 2.71730143368195118202912078686, 4.47250342895832001306731680716, 4.67641402916326046615712643380, 5.43541094583415263882265722168, 6.16492153340069204519733765910, 7.13627283346170109956270670216, 7.930263798021661188105616645105, 8.828599130541840136746916853701