L(s) = 1 | + (15.6 − 9.06i)2-s + (−28.3 + 49.0i)3-s + (100. − 173. i)4-s + (169. + 98.0i)5-s + 1.02e3i·6-s + (−758. + 1.31e3i)7-s − 1.31e3i·8-s + (−509. − 881. i)9-s + 3.55e3·10-s + 5.03e3·11-s + (5.67e3 + 9.83e3i)12-s + (−4.12e3 − 2.38e3i)13-s + 2.74e4i·14-s + (−9.61e3 + 5.55e3i)15-s + (918. + 1.59e3i)16-s + (−2.72e3 + 1.57e3i)17-s + ⋯ |
L(s) = 1 | + (1.38 − 0.801i)2-s + (−0.605 + 1.04i)3-s + (0.783 − 1.35i)4-s + (0.607 + 0.350i)5-s + 1.93i·6-s + (−0.835 + 1.44i)7-s − 0.907i·8-s + (−0.232 − 0.403i)9-s + 1.12·10-s + 1.14·11-s + (0.948 + 1.64i)12-s + (−0.520 − 0.300i)13-s + 2.67i·14-s + (−0.735 + 0.424i)15-s + (0.0560 + 0.0970i)16-s + (−0.134 + 0.0777i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.764i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.645 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.79714 + 1.29902i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.79714 + 1.29902i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (2.74e5 + 1.40e5i)T \) |
good | 2 | \( 1 + (-15.6 + 9.06i)T + (64 - 110. i)T^{2} \) |
| 3 | \( 1 + (28.3 - 49.0i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-169. - 98.0i)T + (3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (758. - 1.31e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 5.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (4.12e3 + 2.38e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (2.72e3 - 1.57e3i)T + (2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-3.75e4 - 2.17e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + 2.30e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 2.95e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.01e5iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (1.50e5 - 2.61e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + 2.77e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.15e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-7.61e5 - 1.31e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.22e5 - 7.06e4i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.44e6 - 1.41e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.36e6 + 2.36e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.93e6 + 3.34e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 3.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (5.07e6 + 2.92e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-2.15e6 - 3.73e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (2.35e6 - 1.35e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.25e7iT - 8.07e13T^{2} \) |
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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
−14.90956384611877730283195611286, −13.79179825890749447271859869357, −12.34280910870731993376131815085, −11.68818428442687645955127556667, −10.29315140559126022552120485676, −9.380751425953950952360201955589, −6.11598645829095297642746623716, −5.37545963418662835390807631507, −3.82118214569225718968583093069, −2.38367360777969318849115258828,
1.02182810082261173526448395097, 3.72832632695602340741500694432, 5.37269616797780184219156384701, 6.84334065555412884019470411906, 7.04868798829100258783073170631, 9.664855278856371086427048310274, 11.68915059915396870656194846929, 12.72224762834640955830064453258, 13.57218462946405831456200169543, 14.15026262048355501692786783451