L(s) = 1 | − 15.4·2-s + 15.5·3-s + 173.·4-s + 161. i·5-s − 240.·6-s + 585. i·7-s − 1.68e3·8-s + 243·9-s − 2.48e3i·10-s + 2.12e3i·11-s + 2.70e3·12-s + 1.92e3·13-s − 9.02e3i·14-s + 2.51e3i·15-s + 1.48e4·16-s − 1.37e3i·17-s + ⋯ |
L(s) = 1 | − 1.92·2-s + 0.577·3-s + 2.70·4-s + 1.29i·5-s − 1.11·6-s + 1.70i·7-s − 3.29·8-s + 0.333·9-s − 2.48i·10-s + 1.59i·11-s + 1.56·12-s + 0.877·13-s − 3.28i·14-s + 0.745i·15-s + 3.62·16-s − 0.279i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.186958 + 0.797897i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.186958 + 0.797897i\) |
\(L(4)\) |
|
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 | \( 1 + (1.09e4 + 5.40e3i)T \) |
good | 2 | \( 1 + 15.4T + 64T^{2} \) |
| 5 | \( 1 - 161. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 585. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 2.12e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.92e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 1.37e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.65e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 - 6.03e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.25e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.55e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.01e5T + 4.75e9T^{2} \) |
| 43 | \( 1 + 8.34e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.19e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 8.70e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.12e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.98e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 2.34e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 7.52e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.39e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 2.21e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.34e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.22e6iT - 8.32e11T^{2} \) |
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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
−14.62417382162482281392977020841, −12.36831204092008798283776443026, −11.38325620631051225113738797094, −10.22597618182687667169246358921, −9.363263937748726245317949112854, −8.399940536197413800965153417901, −7.23207082190165384738451679175, −6.23580871758574956965741137388, −2.77561094875409027891634945706, −2.00338571873144218187945990322,
0.59971014002545584907832687671, 1.34826972373617754064105510775, 3.64239725016103805085960465465, 6.23741022730675300613208375820, 7.82072633889347048183404476009, 8.348302091249288149160693302715, 9.380594864509353837820063506476, 10.47572657588083669654001534275, 11.37407726256460666013506420233, 12.99974213129508375253808590770