L(s) = 1 | + (−0.642 + 0.766i)3-s + (−0.173 − 0.984i)9-s + (−0.524 − 0.439i)11-s + (1.70 − 0.984i)17-s + (1.32 + 0.766i)19-s + (−0.173 + 0.984i)25-s + (0.866 + 0.500i)27-s + (0.673 − 0.118i)33-s + (1.26 − 0.223i)41-s + (−0.223 + 0.266i)43-s + (−0.766 + 0.642i)49-s + (−0.342 + 1.93i)51-s + (−1.43 + 0.524i)57-s + (−0.984 + 0.826i)59-s + (1.85 − 0.326i)67-s + ⋯ |
L(s) = 1 | + (−0.642 + 0.766i)3-s + (−0.173 − 0.984i)9-s + (−0.524 − 0.439i)11-s + (1.70 − 0.984i)17-s + (1.32 + 0.766i)19-s + (−0.173 + 0.984i)25-s + (0.866 + 0.500i)27-s + (0.673 − 0.118i)33-s + (1.26 − 0.223i)41-s + (−0.223 + 0.266i)43-s + (−0.766 + 0.642i)49-s + (−0.342 + 1.93i)51-s + (−1.43 + 0.524i)57-s + (−0.984 + 0.826i)59-s + (1.85 − 0.326i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9342161723\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9342161723\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.642 - 0.766i)T \) |
good | 5 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 11 | \( 1 + (0.524 + 0.439i)T + (0.173 + 0.984i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.223 - 0.266i)T + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.984 - 0.826i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-1.85 + 0.326i)T + (0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.300 + 1.70i)T + (-0.939 - 0.342i)T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \) |
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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
−9.700952093064036125210771423333, −9.060364307554250184372851985216, −7.85471888203819911876098176218, −7.34808845477132884640635570449, −6.05336896393545518115652649418, −5.49562931204940494862571743476, −4.84767230632929533451275520069, −3.59754976808402774211395233801, −3.01352297380702664036971146317, −1.09780757991249140510811360944,
1.05996432983675071744677218802, 2.29222787398104998103519796997, 3.41651496227978527538952999541, 4.74076976142338638934557363342, 5.47437013714142149739416845653, 6.18568881495580892651374769298, 7.13173535479334439417886911267, 7.78523986060443645774560540254, 8.359443347877722930542853394991, 9.659617953345784339905465618535