L(s) = 1 | + (0.866 + 1.5i)2-s + (−1 + 1.73i)4-s − 1.73·8-s + 1.73·11-s + (−0.5 − 0.866i)16-s + (1.49 + 2.59i)22-s + 25-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s + (−1.73 + 3i)44-s + (0.866 + 1.5i)50-s + (−0.866 − 1.5i)53-s − 0.999·64-s + (0.5 − 0.866i)67-s − 1.73·71-s + ⋯ |
L(s) = 1 | + (0.866 + 1.5i)2-s + (−1 + 1.73i)4-s − 1.73·8-s + 1.73·11-s + (−0.5 − 0.866i)16-s + (1.49 + 2.59i)22-s + 25-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s + (−1.73 + 3i)44-s + (0.866 + 1.5i)50-s + (−0.866 − 1.5i)53-s − 0.999·64-s + (0.5 − 0.866i)67-s − 1.73·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.152563084\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.152563084\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.679129110673845028117124772280, −8.050007646720257683899426480233, −7.17344949809127385697036690428, −6.57136521199666472721892390539, −6.22526766865111405430422014758, −5.20550860413768357451081326286, −4.60022366455408694744054172487, −3.82821516926686114720949338764, −3.11552220910246532381051261252, −1.49302199674201314704712759050,
1.06746193244147675682369422061, 1.87321597079727962376842263858, 2.91401199281911393226795243872, 3.68847735340651066749710323682, 4.28138057149627682053376202321, 5.03973262411922178497468603429, 5.92008926471537642011629601388, 6.66001533961012047377803885327, 7.53008405862107227799423092774, 8.911619238861810031072564246896