L(s) = 1 | + (0.456 + 1.33i)2-s + (0.784 + 1.54i)3-s + (−1.58 + 1.22i)4-s + 1.42i·5-s + (−1.70 + 1.75i)6-s − 1.12i·7-s + (−2.35 − 1.56i)8-s + (−1.77 + 2.42i)9-s + (−1.90 + 0.649i)10-s − 1.68·11-s + (−3.12 − 1.48i)12-s + 5.14·13-s + (1.50 − 0.513i)14-s + (−2.19 + 1.11i)15-s + (1.01 − 3.86i)16-s + 4.65i·17-s + ⋯ |
L(s) = 1 | + (0.322 + 0.946i)2-s + (0.452 + 0.891i)3-s + (−0.791 + 0.610i)4-s + 0.636i·5-s + (−0.697 + 0.716i)6-s − 0.425i·7-s + (−0.833 − 0.552i)8-s + (−0.590 + 0.807i)9-s + (−0.602 + 0.205i)10-s − 0.507·11-s + (−0.903 − 0.429i)12-s + 1.42·13-s + (0.402 − 0.137i)14-s + (−0.567 + 0.288i)15-s + (0.253 − 0.967i)16-s + 1.12i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.332031 + 1.47128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.332031 + 1.47128i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.456 - 1.33i)T \) |
| 3 | \( 1 + (-0.784 - 1.54i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 1.42iT - 5T^{2} \) |
| 7 | \( 1 + 1.12iT - 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 - 5.14T + 13T^{2} \) |
| 17 | \( 1 - 4.65iT - 17T^{2} \) |
| 19 | \( 1 + 5.86iT - 19T^{2} \) |
| 29 | \( 1 - 3.67iT - 29T^{2} \) |
| 31 | \( 1 - 3.79iT - 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 + 8.23iT - 41T^{2} \) |
| 43 | \( 1 - 3.72iT - 43T^{2} \) |
| 47 | \( 1 - 7.62T + 47T^{2} \) |
| 53 | \( 1 - 2.23iT - 53T^{2} \) |
| 59 | \( 1 + 3.98T + 59T^{2} \) |
| 61 | \( 1 + 7.14T + 61T^{2} \) |
| 67 | \( 1 + 12.2iT - 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 5.27iT - 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 15.9iT - 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−12.59160594779578627723783646075, −10.91625615680669746790203431180, −10.56331852362145123256496323738, −9.110250651599318723483009568576, −8.506101392953300059466822132504, −7.38673198622285837279952498238, −6.33460157521210903508372287213, −5.16778514105703081715043116418, −4.00924063811188230227536804015, −3.07636280201761253392868984561,
1.12398455331080918405575204036, 2.54489885727019502804143570611, 3.80786460576964790088184820872, 5.32310696033698963905360964733, 6.25225234331437271372173991304, 7.891625037309704357315182556191, 8.710027004353670164669201386854, 9.467230885273615173331933464547, 10.74324149615978363678462274763, 11.78725158564962132850121954422