L(s) = 1 | + 1.50i·2-s + (−1.53 − 0.796i)3-s − 0.267·4-s + (1.12 − 1.93i)5-s + (1.19 − 2.31i)6-s + 2.12·7-s + 2.60i·8-s + (1.73 + 2.44i)9-s + (2.90 + 1.69i)10-s + (3.27 − 0.517i)11-s + (0.412 + 0.213i)12-s − 2.12·13-s + 3.20i·14-s + (−3.27 + 2.07i)15-s − 4.46·16-s − 4.11i·17-s + ⋯ |
L(s) = 1 | + 1.06i·2-s + (−0.888 − 0.459i)3-s − 0.133·4-s + (0.503 − 0.863i)5-s + (0.489 − 0.945i)6-s + 0.804·7-s + 0.922i·8-s + (0.577 + 0.816i)9-s + (0.920 + 0.536i)10-s + (0.987 − 0.156i)11-s + (0.118 + 0.0615i)12-s − 0.590·13-s + 0.857i·14-s + (−0.844 + 0.535i)15-s − 1.11·16-s − 0.997i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04786 + 0.395687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04786 + 0.395687i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.53 + 0.796i)T \) |
| 5 | \( 1 + (-1.12 + 1.93i)T \) |
| 11 | \( 1 + (-3.27 + 0.517i)T \) |
good | 2 | \( 1 - 1.50iT - 2T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 17 | \( 1 + 4.11iT - 17T^{2} \) |
| 19 | \( 1 - 4.63iT - 19T^{2} \) |
| 23 | \( 1 - 5.32T + 23T^{2} \) |
| 29 | \( 1 + 8.95T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 1.16iT - 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 + 9.50T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + 4.50T + 53T^{2} \) |
| 59 | \( 1 + 4.89iT - 59T^{2} \) |
| 61 | \( 1 + 3.39iT - 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 - 13.3iT - 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 + 13.8iT - 79T^{2} \) |
| 83 | \( 1 - 9.33iT - 83T^{2} \) |
| 89 | \( 1 + 4.62iT - 89T^{2} \) |
| 97 | \( 1 + 1.16iT - 97T^{2} \) |
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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
−12.99660702388979895521883644138, −11.83540314839012222404390656643, −11.31818971079687013975171882668, −9.780109723815690851297448059737, −8.533102051154467395035124574664, −7.52945227844099289308631861419, −6.54861361834886718282764938872, −5.47173169179997644823605480088, −4.78040959710920070437530203608, −1.67655030649370817308740798847,
1.72982313597928638553338686196, 3.45652697805316768804354428840, 4.81027770908956435359586810080, 6.31444066958710588079820270878, 7.16828426624672041807152228871, 9.216765377664647021641964187086, 10.03464609235382953090693360086, 11.05044862788892748718786031323, 11.32430720068435844536970748218, 12.35065273610747886515310251053