L(s) = 1 | + (0.866 − 0.5i)2-s + (−1.67 − 0.448i)3-s + (0.499 − 0.866i)4-s + (−0.258 − 0.965i)5-s + (−1.67 + 0.448i)6-s + (−0.5 + 0.866i)7-s − 0.999i·8-s + (1.73 + 1.00i)9-s + (−0.707 − 0.707i)10-s + (−1.22 + 1.22i)12-s + (0.258 + 0.965i)13-s + 0.999i·14-s + 1.73i·15-s + (−0.5 − 0.866i)16-s + 2·18-s + (0.517 + 1.93i)19-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (−1.67 − 0.448i)3-s + (0.499 − 0.866i)4-s + (−0.258 − 0.965i)5-s + (−1.67 + 0.448i)6-s + (−0.5 + 0.866i)7-s − 0.999i·8-s + (1.73 + 1.00i)9-s + (−0.707 − 0.707i)10-s + (−1.22 + 1.22i)12-s + (0.258 + 0.965i)13-s + 0.999i·14-s + 1.73i·15-s + (−0.5 − 0.866i)16-s + 2·18-s + (0.517 + 1.93i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8412106475\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8412106475\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.258 + 0.965i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.258 - 0.965i)T \) |
good | 3 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 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
−8.874301170263662061298555297714, −7.72065031136666514876823942079, −6.97359449463822830351270606524, −5.93057151638271799500203706084, −5.81517080057199859857016584623, −5.15543867817585582569059574687, −4.24752221621575366430303258872, −3.52581366116622463503774758500, −1.88134356204523515300022254660, −1.28605414627110543921021063043,
0.48372960094865830063581300922, 2.65386928807635832853550181024, 3.49679011069362165810620173664, 4.36519001604430602625039597487, 4.89012379831672919320329967257, 5.83318466519566916691429966185, 6.41269976363302089314272583776, 6.92879437021257041616517061878, 7.45862586034145182482687611858, 8.500998185487629683329744617920