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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.431389117\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.431389117\) |
\(L(1)\) |
|
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
−9.322730816874255052951062931267, −8.599060599155582273141270236579, −7.67206466056263924884794365459, −7.17905323628529530361231066997, −6.33822972631082758716890936656, −5.41761277491083940750928498418, −4.26757656935865196033526406812, −3.28562543631308725523060099537, −2.35881529164247307111705475122, −1.17056272681199540859902103878,
1.66702482944725020392179195104, 2.91038200973107212401893996476, 3.82314432191690581344671909579, 4.42304988613756041416467245716, 5.66990910050543165968201883013, 6.22447884678874661299875171559, 7.54012641307481176310182504114, 8.245855980504436023488328171028, 8.742147619838937066992244951794, 9.731850186834608609525283991710