L(s) = 1 | + (0.961 − 1.75i)2-s + (−2.86 − 0.903i)3-s + (−2.15 − 3.37i)4-s + (−4.95 − 0.663i)5-s + (−4.33 + 4.14i)6-s + (7.30 − 7.30i)7-s + (−7.98 + 0.535i)8-s + (7.36 + 5.17i)9-s + (−5.92 + 8.05i)10-s + 4.41·11-s + (3.11 + 11.5i)12-s + (7.53 − 7.53i)13-s + (−5.78 − 19.8i)14-s + (13.5 + 6.37i)15-s + (−6.73 + 14.5i)16-s + (0.350 − 0.350i)17-s + ⋯ |
L(s) = 1 | + (0.480 − 0.876i)2-s + (−0.953 − 0.301i)3-s + (−0.538 − 0.842i)4-s + (−0.991 − 0.132i)5-s + (−0.722 + 0.691i)6-s + (1.04 − 1.04i)7-s + (−0.997 + 0.0669i)8-s + (0.818 + 0.574i)9-s + (−0.592 + 0.805i)10-s + 0.401·11-s + (0.259 + 0.965i)12-s + (0.579 − 0.579i)13-s + (−0.413 − 1.41i)14-s + (0.905 + 0.425i)15-s + (−0.420 + 0.907i)16-s + (0.0206 − 0.0206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.630i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.323589 - 0.910789i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.323589 - 0.910789i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.961 + 1.75i)T \) |
| 3 | \( 1 + (2.86 + 0.903i)T \) |
| 5 | \( 1 + (4.95 + 0.663i)T \) |
good | 7 | \( 1 + (-7.30 + 7.30i)T - 49iT^{2} \) |
| 11 | \( 1 - 4.41T + 121T^{2} \) |
| 13 | \( 1 + (-7.53 + 7.53i)T - 169iT^{2} \) |
| 17 | \( 1 + (-0.350 + 0.350i)T - 289iT^{2} \) |
| 19 | \( 1 + 9.24T + 361T^{2} \) |
| 23 | \( 1 + (-17.9 + 17.9i)T - 529iT^{2} \) |
| 29 | \( 1 + 5.52T + 841T^{2} \) |
| 31 | \( 1 - 48.1iT - 961T^{2} \) |
| 37 | \( 1 + (-3.39 - 3.39i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 33.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (1.45 + 1.45i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (27.8 + 27.8i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-52.6 - 52.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 24.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-32.1 + 32.1i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 116.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (72.2 - 72.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 55.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + (46.5 - 46.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 33.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (24.6 + 24.6i)T + 9.40e3iT^{2} \) |
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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
−14.20184962365037554390114725299, −12.99176699566604787623787771536, −12.03185207194277517139881084670, −11.03380624484955893038768489254, −10.54787436595850520141641379990, −8.424553144121780073654156133765, −6.92382721982920864111743122738, −5.08481264900367531848336599726, −4.00681070155977037649204068598, −1.00524080701625502659281395825,
4.04216664351720539135055381288, 5.22612425576941844719276778319, 6.53836259289736088847386609726, 7.942298375922568194923053255401, 9.143153703187121080686710411539, 11.35086627827089545791657982757, 11.71963113475268774759636761498, 12.96194430128162916912484506093, 14.69330585124272592499546310479, 15.25060181129013863176787826733