L(s) = 1 | + 1.75i·2-s + (−1.12 + 1.31i)3-s − 1.09·4-s + (0.5 − 0.866i)5-s + (−2.31 − 1.98i)6-s + (−1.42 + 2.22i)7-s + 1.59i·8-s + (−0.458 − 2.96i)9-s + (1.52 + 0.879i)10-s + (−2.01 + 1.16i)11-s + (1.23 − 1.43i)12-s + (−4.66 + 2.69i)13-s + (−3.91 − 2.50i)14-s + (0.575 + 1.63i)15-s − 4.99·16-s + (2.98 − 5.17i)17-s + ⋯ |
L(s) = 1 | + 1.24i·2-s + (−0.650 + 0.759i)3-s − 0.546·4-s + (0.223 − 0.387i)5-s + (−0.944 − 0.809i)6-s + (−0.538 + 0.842i)7-s + 0.564i·8-s + (−0.152 − 0.988i)9-s + (0.481 + 0.278i)10-s + (−0.606 + 0.350i)11-s + (0.355 − 0.414i)12-s + (−1.29 + 0.746i)13-s + (−1.04 − 0.670i)14-s + (0.148 + 0.421i)15-s − 1.24·16-s + (0.725 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.224650 - 0.747594i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.224650 - 0.747594i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.12 - 1.31i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.42 - 2.22i)T \) |
good | 2 | \( 1 - 1.75iT - 2T^{2} \) |
| 11 | \( 1 + (2.01 - 1.16i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.66 - 2.69i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.98 + 5.17i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.71 - 0.992i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.11 - 2.37i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.23 - 3.59i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.61iT - 31T^{2} \) |
| 37 | \( 1 + (-3.01 - 5.22i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.20 + 2.08i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.02 - 10.4i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 7.29T + 47T^{2} \) |
| 53 | \( 1 + (1.49 + 0.864i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 8.20T + 59T^{2} \) |
| 61 | \( 1 - 8.24iT - 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 12.8iT - 71T^{2} \) |
| 73 | \( 1 + (3.50 + 2.02i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + (-2.01 + 3.49i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.0896 - 0.155i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.78 - 3.91i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.06820170411627832916798065169, −11.46862130445812597856730375625, −9.954976636326651686122401874132, −9.451622990648847520701362681361, −8.411872791805702322993088545929, −7.14149442654894136400558096538, −6.34585191435738550357889433584, −5.16183506433717938625415549964, −4.89093633112005974864083866110, −2.76669764372770068182819764888,
0.58189119131004099750928266365, 2.25472408778471599454517297466, 3.35483301806500731812004319773, 4.92369243904108030577974169750, 6.29647357583058290082811665968, 7.12618682518332954203617324301, 8.157208128949220663160666299972, 9.893628911081358660511595847550, 10.42667174139224290142175607550, 10.97123021398173543914130351413