L(s) = 1 | + 13.3·2-s − 15.5·3-s + 114.·4-s + 40.7i·5-s − 208.·6-s + 469. i·7-s + 675.·8-s + 243·9-s + 544. i·10-s + 1.56e3i·11-s − 1.78e3·12-s + 3.79e3·13-s + 6.27e3i·14-s − 635. i·15-s + 1.69e3·16-s − 9.02e3i·17-s + ⋯ |
L(s) = 1 | + 1.67·2-s − 0.577·3-s + 1.78·4-s + 0.326i·5-s − 0.964·6-s + 1.36i·7-s + 1.31·8-s + 0.333·9-s + 0.544i·10-s + 1.17i·11-s − 1.03·12-s + 1.72·13-s + 2.28i·14-s − 0.188i·15-s + 0.413·16-s − 1.83i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(3.44256 + 2.00551i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.44256 + 2.00551i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 15.5T \) |
| 23 | \( 1 + (6.00e3 - 1.05e4i)T \) |
good | 2 | \( 1 - 13.3T + 64T^{2} \) |
| 5 | \( 1 - 40.7iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 469. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.56e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.79e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 9.02e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 7.10e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 3.65e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.75e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 7.51e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.16e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + 7.70e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.28e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.49e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 5.59e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.67e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.99e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.20e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 6.59e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.58e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 2.93e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 4.30e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.56e5iT - 8.32e11T^{2} \) |
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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
−13.65506581632664842412341738261, −12.50622632437103251920455526416, −11.89015760192043964355339228446, −10.92720895233970921909925710536, −9.213126536517667681675381987691, −7.18618487164893973366640314838, −5.95293428725246473208849145275, −5.18946320133748025175708179255, −3.66714432920225969930290966284, −2.12557648180179629227571092154,
1.07131371376534420386286105074, 3.52517052029187335447792193180, 4.39155120708257403181402409360, 5.88324500626046990538626125985, 6.66061642999113385954422082868, 8.436767992226026046500850750239, 10.81190249253243351007543711143, 11.04040364675975881763374935890, 12.62643835188365202575382421656, 13.35152465184363432021123549269