L(s) = 1 | + 2-s + (−0.396 − 1.68i)3-s + 4-s + 2.37i·5-s + (−0.396 − 1.68i)6-s + (−0.792 + 2.52i)7-s + 8-s + (−2.68 + 1.33i)9-s + 2.37i·10-s + 5.74·11-s + (−0.396 − 1.68i)12-s + (3.46 − i)13-s + (−0.792 + 2.52i)14-s + (4 − 0.939i)15-s + 16-s + 2.52·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.228 − 0.973i)3-s + 0.5·4-s + 1.06i·5-s + (−0.161 − 0.688i)6-s + (−0.299 + 0.954i)7-s + 0.353·8-s + (−0.895 + 0.445i)9-s + 0.750i·10-s + 1.73·11-s + (−0.114 − 0.486i)12-s + (0.960 − 0.277i)13-s + (−0.211 + 0.674i)14-s + (1.03 − 0.242i)15-s + 0.250·16-s + 0.612·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.11171 + 0.220059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11171 + 0.220059i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.396 + 1.68i)T \) |
| 7 | \( 1 + (0.792 - 2.52i)T \) |
| 13 | \( 1 + (-3.46 + i)T \) |
good | 5 | \( 1 - 2.37iT - 5T^{2} \) |
| 11 | \( 1 - 5.74T + 11T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 + 7.57T + 19T^{2} \) |
| 23 | \( 1 - 6.78iT - 23T^{2} \) |
| 29 | \( 1 + 7.57iT - 29T^{2} \) |
| 31 | \( 1 - 5.84T + 31T^{2} \) |
| 37 | \( 1 + 4.90iT - 37T^{2} \) |
| 41 | \( 1 - 2.62iT - 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 + 4.62iT - 47T^{2} \) |
| 53 | \( 1 + 3.16iT - 53T^{2} \) |
| 59 | \( 1 - 8.74iT - 59T^{2} \) |
| 61 | \( 1 + 3.74iT - 61T^{2} \) |
| 67 | \( 1 + 2.67iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 3.61T + 73T^{2} \) |
| 79 | \( 1 + 9.37T + 79T^{2} \) |
| 83 | \( 1 - 4.74iT - 83T^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 - 7.42T + 97T^{2} \) |
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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
−11.32629795697688771793270304183, −10.15591631376283723663654045992, −8.910304524392227918526947764830, −8.002180395315577530520934771969, −6.73685577514539599707686975707, −6.36421149379313255562045509688, −5.65445045847651439539508044947, −3.93050173832710218836571642714, −2.90725762617720956931240085978, −1.71162671381079151204383656415,
1.19691416409276258089308247251, 3.41686910836999343846367094809, 4.26526917236577712319365534431, 4.73322594685884262515154529274, 6.20374911858328709146765019492, 6.66306448925258549295576919409, 8.474582967254196786440475161920, 8.907535059574882873704791638157, 10.07707509770341978662492389811, 10.77266252891494870073356564992