L(s) = 1 | + (−0.618 − 1.61i)3-s + 5-s + (2.61 + 0.381i)7-s + (−2.23 + 2.00i)9-s − 5.23i·11-s + 3.23i·13-s + (−0.618 − 1.61i)15-s + 4.47·17-s + 2.76i·19-s + (−1.00 − 4.47i)21-s − 5.70i·23-s + 25-s + (4.61 + 2.38i)27-s − 4i·29-s + 1.23i·31-s + ⋯ |
L(s) = 1 | + (−0.356 − 0.934i)3-s + 0.447·5-s + (0.989 + 0.144i)7-s + (−0.745 + 0.666i)9-s − 1.57i·11-s + 0.897i·13-s + (−0.159 − 0.417i)15-s + 1.08·17-s + 0.634i·19-s + (−0.218 − 0.975i)21-s − 1.19i·23-s + 0.200·25-s + (0.888 + 0.458i)27-s − 0.742i·29-s + 0.222i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.831540768\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.831540768\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.618 + 1.61i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.61 - 0.381i)T \) |
good | 11 | \( 1 + 5.23iT - 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.76iT - 19T^{2} \) |
| 23 | \( 1 + 5.70iT - 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 1.23iT - 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 + 4.76iT - 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 10.1iT - 71T^{2} \) |
| 73 | \( 1 - 11.2iT - 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 - 3.52T + 89T^{2} \) |
| 97 | \( 1 + 8.18iT - 97T^{2} \) |
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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
−8.915218973697059754957765721384, −8.301768586993001321361759572449, −7.71021958800989852203567538103, −6.67871807071117226569253296551, −5.89684362243154577615975544855, −5.41109133899810486251578748803, −4.24457801198641321980019397214, −2.90675965730011317495914818087, −1.86825343897392893228175158672, −0.851305409811585209475125479449,
1.25920285030274309986933434344, 2.59564926289843490116500942374, 3.76477662353238744186008108730, 4.74577941876334570749897244674, 5.25402116331487917785964787108, 6.03048812070819602853458345429, 7.34039863911765105955861086731, 7.79445766423773592961534822526, 9.055692876166299090584907358832, 9.496207548233981401469113318733