L(s) = 1 | + 1.73·3-s − 3.46i·7-s + 2.99·9-s + 3.46·11-s − 4·13-s − 6i·17-s − 3.46i·19-s − 5.99i·21-s − 3.46·23-s + 5.19·27-s + 6i·29-s + 3.46i·31-s + 5.99·33-s + 4·37-s − 6.92·39-s + ⋯ |
L(s) = 1 | + 1.00·3-s − 1.30i·7-s + 0.999·9-s + 1.04·11-s − 1.10·13-s − 1.45i·17-s − 0.794i·19-s − 1.30i·21-s − 0.722·23-s + 1.00·27-s + 1.11i·29-s + 0.622i·31-s + 1.04·33-s + 0.657·37-s − 1.10·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.326890053\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.326890053\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 12iT - 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.509454266858874177720613326992, −8.972931571660743922204958213263, −7.85498510121115723901522351805, −7.11360322846134384945366638025, −6.78820023125593017989067828946, −5.06278832377350930361764924146, −4.27926499727315289089739939820, −3.42830021632819486182643434622, −2.34015288593240148834715663035, −0.927406560044210912211595088998,
1.74616850029122072339844568918, 2.51901527819013492998931905679, 3.70721354746632601359932475889, 4.51559807789055526208734608411, 5.84430080541105412271687470428, 6.46808189778991609006335920607, 7.76431221982458566106021140880, 8.207093751741031091844427746137, 9.154497285542077631355645209698, 9.617174337322812066821998858002