L(s) = 1 | + (−1.03 + 5.56i)2-s + (13.3 − 8.09i)3-s + (−29.8 − 11.5i)4-s + (55.7 − 4.70i)5-s + (31.2 + 82.4i)6-s − 92.5·7-s + (95.0 − 154. i)8-s + (111. − 215. i)9-s + (−31.6 + 314. i)10-s + 666.·11-s + (−490. + 88.1i)12-s − 717. i·13-s + (95.9 − 514. i)14-s + (703. − 513. i)15-s + (758. + 688. i)16-s + 525.·17-s + ⋯ |
L(s) = 1 | + (−0.183 + 0.983i)2-s + (0.854 − 0.519i)3-s + (−0.932 − 0.360i)4-s + (0.996 − 0.0840i)5-s + (0.354 + 0.935i)6-s − 0.713·7-s + (0.525 − 0.850i)8-s + (0.460 − 0.887i)9-s + (−0.0999 + 0.994i)10-s + 1.66·11-s + (−0.984 + 0.176i)12-s − 1.17i·13-s + (0.130 − 0.701i)14-s + (0.807 − 0.589i)15-s + (0.740 + 0.672i)16-s + 0.440·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.18764 + 0.288140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18764 + 0.288140i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.03 - 5.56i)T \) |
| 3 | \( 1 + (-13.3 + 8.09i)T \) |
| 5 | \( 1 + (-55.7 + 4.70i)T \) |
good | 7 | \( 1 + 92.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 666.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 717. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 525.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.76e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.08e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.49e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.60e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.28e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.26e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.57e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.14e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.48e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.58e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.48e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.20e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.08e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 2.73e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.04e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.59e5iT - 8.58e9T^{2} \) |
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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
−14.25057808054980087359553510370, −13.33964453748427515662154664001, −12.42682766070273061115427061614, −9.951688153390968033439922535718, −9.344010305698017930984948511191, −8.123379405272746481827597416183, −6.75090917263390466054947642053, −5.81510534974904233579782583768, −3.58816245374323162134942676462, −1.27053747639851188164132619900,
1.68772405806866652512773520140, 3.16873583304338851424518913093, 4.56828120236892870059925342867, 6.70498830182050525502851700893, 8.871796085107367261153139033855, 9.351218297028075450760044259822, 10.29880543389801062012225081854, 11.68402883653209799960571582775, 13.06022432152909740014297342070, 13.95521202995896529611732714365