L(s) = 1 | + (1.34 + 0.437i)2-s + (−1.86 + 1.35i)3-s + (1.61 + 1.17i)4-s + (−2.45 − 7.55i)5-s + (−3.10 + 1.00i)6-s + (−2.13 + 2.93i)7-s + (1.66 + 2.28i)8-s + (−1.13 + 3.49i)9-s − 11.2i·10-s + (10.5 + 3.17i)11-s − 4.61·12-s + (−10.5 − 3.41i)13-s + (−4.15 + 3.01i)14-s + (14.8 + 10.7i)15-s + (1.23 + 3.80i)16-s + (20.0 − 6.52i)17-s + ⋯ |
L(s) = 1 | + (0.672 + 0.218i)2-s + (−0.622 + 0.452i)3-s + (0.404 + 0.293i)4-s + (−0.490 − 1.51i)5-s + (−0.517 + 0.168i)6-s + (−0.304 + 0.419i)7-s + (0.207 + 0.286i)8-s + (−0.126 + 0.388i)9-s − 1.12i·10-s + (0.957 + 0.288i)11-s − 0.384·12-s + (−0.807 − 0.262i)13-s + (−0.296 + 0.215i)14-s + (0.988 + 0.718i)15-s + (0.0772 + 0.237i)16-s + (1.18 − 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.984567 + 0.155589i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.984567 + 0.155589i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.34 - 0.437i)T \) |
| 11 | \( 1 + (-10.5 - 3.17i)T \) |
good | 3 | \( 1 + (1.86 - 1.35i)T + (2.78 - 8.55i)T^{2} \) |
| 5 | \( 1 + (2.45 + 7.55i)T + (-20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (2.13 - 2.93i)T + (-15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (10.5 + 3.41i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (-20.0 + 6.52i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-4.09 - 5.63i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + 34.5T + 529T^{2} \) |
| 29 | \( 1 + (-25.8 + 35.5i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (5.73 - 17.6i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-5.08 - 3.69i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (0.251 + 0.346i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + 7.55iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (2.74 - 1.99i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (13.3 - 41.0i)T + (-2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (33.1 + 24.0i)T + (1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-39.9 + 12.9i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + 5.04T + 4.48e3T^{2} \) |
| 71 | \( 1 + (7.75 + 23.8i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (39.9 - 55.0i)T + (-1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-88.6 - 28.7i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (102. - 33.2i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 23.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + (5.59 - 17.2i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−17.27290765805819748698156861490, −16.47188519927107245509731687335, −15.68631551538239391532977925068, −14.05937132342970549726624722996, −12.34838081338113067558178066622, −11.89407768705391552435326262790, −9.762117754414821985974086707540, −8.009644472684696138628166266376, −5.66885838076168589933895992365, −4.43574587513077144382252248678,
3.52873176118917844243502079164, 6.21795301850528591583957092807, 7.22680654322096989191572480543, 10.11532752573165730307077563527, 11.46291522130423817701406505433, 12.26329128327609799785249811071, 14.13882706001167314687756416111, 14.83322321482655235168725507232, 16.49168369350648899907168607802, 17.89422968160770328163230420296