L(s) = 1 | + (−0.5 + 0.866i)3-s − 7-s + (−0.499 − 0.866i)9-s + 2·11-s + (1.5 + 2.59i)13-s + (−2 + 3.46i)17-s + (−4 − 1.73i)19-s + (0.5 − 0.866i)21-s + (2 + 3.46i)23-s + (2.5 + 4.33i)25-s + 0.999·27-s + 3·31-s + (−1 + 1.73i)33-s − 5·37-s − 3·39-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s − 0.377·7-s + (−0.166 − 0.288i)9-s + 0.603·11-s + (0.416 + 0.720i)13-s + (−0.485 + 0.840i)17-s + (−0.917 − 0.397i)19-s + (0.109 − 0.188i)21-s + (0.417 + 0.722i)23-s + (0.5 + 0.866i)25-s + 0.192·27-s + 0.538·31-s + (−0.174 + 0.301i)33-s − 0.821·37-s − 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.626322 + 0.874553i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.626322 + 0.874553i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + (2 - 3.46i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 - 7.79i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7 - 12.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7 + 12.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.47140252490331208747956144629, −9.296795405918880048668634561667, −9.021324787538220114908623211405, −7.87839890661223125054854045295, −6.61165737228390701623702765014, −6.25237145859147901907165990483, −4.94932860178085626855492071007, −4.11601781371529114005569547984, −3.14962463111515055093167669483, −1.54886931932192174762563354619,
0.54194631215494794124172825010, 2.13091480285622525382509842280, 3.34551188670806943062898976178, 4.53488921649565395276253547459, 5.57814408467689425737300996369, 6.57377424807135374838341145114, 7.01332594763960408665411084164, 8.329606155249876670375813681076, 8.776645008710452617809812922838, 9.993617443608869853442800013521