L(s) = 1 | + (0.866 − 1.5i)3-s + (2.09 − 3.63i)5-s − 4.28·7-s − 4.71·11-s + (−0.615 − 1.06i)13-s + (−3.63 − 6.29i)15-s + (−1.61 + 2.79i)17-s + (0.625 + 4.31i)19-s + (−3.71 + 6.43i)21-s + (−1.90 − 3.29i)23-s + (−6.31 − 10.9i)25-s + 5.19·27-s + (−2.09 − 3.63i)29-s − 0.825·31-s + (−4.08 + 7.07i)33-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)3-s + (0.938 − 1.62i)5-s − 1.62·7-s − 1.42·11-s + (−0.170 − 0.295i)13-s + (−0.938 − 1.62i)15-s + (−0.391 + 0.678i)17-s + (0.143 + 0.989i)19-s + (−0.810 + 1.40i)21-s + (−0.397 − 0.687i)23-s + (−1.26 − 2.18i)25-s + 1.00·27-s + (−0.389 − 0.675i)29-s − 0.148·31-s + (−0.710 + 1.23i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.058452922\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.058452922\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.625 - 4.31i)T \) |
good | 3 | \( 1 + (-0.866 + 1.5i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.09 + 3.63i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 4.28T + 7T^{2} \) |
| 11 | \( 1 + 4.71T + 11T^{2} \) |
| 13 | \( 1 + (0.615 + 1.06i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.61 - 2.79i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.90 + 3.29i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.09 + 3.63i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 0.825T + 31T^{2} \) |
| 37 | \( 1 + 2.96T + 37T^{2} \) |
| 41 | \( 1 + (-2.69 + 4.67i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.49 + 2.58i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.653 - 1.13i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.38 + 2.39i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.31 + 12.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.09 - 7.10i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.69 + 2.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.04 + 7.01i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.69 - 8.13i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.31 - 4.01i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.32T + 83T^{2} \) |
| 89 | \( 1 + (5.61 + 9.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.92 - 10.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.268327700068222756174546745347, −8.386809302937897296451858284036, −7.914769939001312206547620408240, −6.77143663478275602818584074227, −5.87924846444489146873680998847, −5.28389766850388976463619635016, −4.03629063757321649267407055259, −2.64222186106809838964356152539, −1.85435982647179703793754547193, −0.38054409505408447381049380903,
2.53988423264868888707884270399, 2.91838013073248156331484209828, 3.73976606857255447688193562096, 5.14694192971919266412731108012, 6.09372152946337469559567192571, 6.87439208816188214049259356223, 7.40934235336269704155056115732, 8.964414284268660753696017174154, 9.596113323848815057768461902978, 10.01370446681001925741668722399