L(s) = 1 | + (−0.5 − 0.866i)2-s + (−1.5 − 0.866i)3-s + (0.500 − 0.866i)4-s − 5-s + 1.73i·6-s + (0.5 − 2.59i)7-s − 3·8-s + (1.5 + 2.59i)9-s + (0.5 + 0.866i)10-s + 5·11-s + (−1.5 + 0.866i)12-s + (2.5 + 4.33i)13-s + (−2.5 + 0.866i)14-s + (1.5 + 0.866i)15-s + (0.500 + 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.866 − 0.499i)3-s + (0.250 − 0.433i)4-s − 0.447·5-s + 0.707i·6-s + (0.188 − 0.981i)7-s − 1.06·8-s + (0.5 + 0.866i)9-s + (0.158 + 0.273i)10-s + 1.50·11-s + (−0.433 + 0.249i)12-s + (0.693 + 1.20i)13-s + (−0.668 + 0.231i)14-s + (0.387 + 0.223i)15-s + (0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.368746 - 0.500724i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.368746 - 0.500724i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.51799207339761565762672581044, −13.46385107895923437615042734149, −11.71708196006409488967224957758, −11.56480729303810202545692114285, −10.39171380354547441234432551552, −9.058662928250434748634489185049, −7.16140752125138540149038860546, −6.25357711914196792490783242610, −4.24046345121247578862061925991, −1.34481939087700784078345092839,
3.72236946875674149886478688528, 5.69288996773464311684903806541, 6.71366573090050386876404957274, 8.316946475014102569650881926035, 9.293965850403350358128547671807, 11.02430094162689954314665678942, 11.86846019163676428235716632122, 12.70978823332168147455043032808, 14.87376368474501683723996129600, 15.47674288230236081681088806183