L(s) = 1 | + 2.45·2-s + (−1.72 + 0.157i)3-s + 4.01·4-s + 3.63i·5-s + (−4.22 + 0.385i)6-s + i·7-s + 4.92·8-s + (2.95 − 0.542i)9-s + 8.91i·10-s + (−1.88 − 2.72i)11-s + (−6.91 + 0.630i)12-s − 5.76i·13-s + 2.45i·14-s + (−0.571 − 6.27i)15-s + 4.06·16-s + 4.19·17-s + ⋯ |
L(s) = 1 | + 1.73·2-s + (−0.995 + 0.0907i)3-s + 2.00·4-s + 1.62i·5-s + (−1.72 + 0.157i)6-s + 0.377i·7-s + 1.74·8-s + (0.983 − 0.180i)9-s + 2.81i·10-s + (−0.568 − 0.822i)11-s + (−1.99 + 0.181i)12-s − 1.59i·13-s + 0.655i·14-s + (−0.147 − 1.61i)15-s + 1.01·16-s + 1.01·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.18530 + 0.791566i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18530 + 0.791566i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.72 - 0.157i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (1.88 + 2.72i)T \) |
good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 5 | \( 1 - 3.63iT - 5T^{2} \) |
| 13 | \( 1 + 5.76iT - 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 2.05iT - 19T^{2} \) |
| 23 | \( 1 - 3.24iT - 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 + 0.628T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 - 0.462iT - 43T^{2} \) |
| 47 | \( 1 + 4.50iT - 47T^{2} \) |
| 53 | \( 1 - 1.38iT - 53T^{2} \) |
| 59 | \( 1 - 0.774iT - 59T^{2} \) |
| 61 | \( 1 + 5.90iT - 61T^{2} \) |
| 67 | \( 1 + 0.632T + 67T^{2} \) |
| 71 | \( 1 - 9.61iT - 71T^{2} \) |
| 73 | \( 1 - 14.3iT - 73T^{2} \) |
| 79 | \( 1 + 7.56iT - 79T^{2} \) |
| 83 | \( 1 - 6.21T + 83T^{2} \) |
| 89 | \( 1 - 5.60iT - 89T^{2} \) |
| 97 | \( 1 + 8.98T + 97T^{2} \) |
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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
−12.38388664389573494184093784776, −11.40473622877074011350893016931, −10.84254045764394072597471569749, −10.10533058846905708928942962962, −7.72340695720564725948494148622, −6.78756780024381109920599638650, −5.76745314509541733729376881033, −5.34952614481961043884746521745, −3.59441097309565515358261581276, −2.80439336392556313407222723438,
1.69939978010165784888397619548, 4.10254479858703210372091671226, 4.72900564122502442347292085976, 5.48723617420018659309219799393, 6.58704681180218379459053514404, 7.67833054532870463811723269328, 9.343169225200480751170165319958, 10.56150593361501225637743840132, 11.77421256670448954631770466377, 12.29017802612851438583927108220