L(s) = 1 | + 10.2i·2-s + 141. i·3-s + 151.·4-s − 508. i·5-s − 1.44e3·6-s − 610. i·7-s + 4.16e3i·8-s − 1.33e4·9-s + 5.19e3·10-s − 2.33e4·11-s + 2.13e4i·12-s − 3.82e4·13-s + 6.24e3·14-s + 7.17e4·15-s − 3.80e3·16-s + 8.27e3·17-s + ⋯ |
L(s) = 1 | + 0.638i·2-s + 1.74i·3-s + 0.591·4-s − 0.813i·5-s − 1.11·6-s − 0.254i·7-s + 1.01i·8-s − 2.03·9-s + 0.519·10-s − 1.59·11-s + 1.03i·12-s − 1.33·13-s + 0.162·14-s + 1.41·15-s − 0.0580·16-s + 0.0990·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.423560 - 0.764295i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.423560 - 0.764295i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (1.81e6 - 2.89e6i)T \) |
good | 2 | \( 1 - 10.2iT - 256T^{2} \) |
| 3 | \( 1 - 141. iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 508. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 610. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 2.33e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 3.82e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 8.27e3T + 6.97e9T^{2} \) |
| 19 | \( 1 + 4.39e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 1.09e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 1.33e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 6.24e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.28e4iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 4.22e6T + 7.98e12T^{2} \) |
| 47 | \( 1 - 1.43e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.03e7T + 6.22e13T^{2} \) |
| 59 | \( 1 - 8.60e6T + 1.46e14T^{2} \) |
| 61 | \( 1 - 1.46e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 9.05e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.39e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.77e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 2.98e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 3.68e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 9.08e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 4.00e7T + 7.83e15T^{2} \) |
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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
−15.23333521698763959390543651022, −14.40811185408229159597203039582, −12.58386785073759647410392147852, −11.03336289568711984652611126950, −10.17735149273073569412855917748, −8.879920570656260279259065005712, −7.55593273744680749948792987093, −5.43498573743155186660501214486, −4.76365698932405792987668634944, −2.79824140741035278182921035090,
0.29083841340393805562481991794, 2.15108728313600920306523138924, 2.73377352359813864693730188404, 5.87764687332680581063221840703, 7.16264203701331887058551454219, 7.82842661542077392135806109782, 10.11163019203987001656966449554, 11.30303635455797897735088707546, 12.31654333648049722171807011766, 13.02881980784803663479731665164