L(s) = 1 | + (−2.14 + 1.55i)2-s + (1.55 − 4.78i)4-s + (1.38 + 1.00i)5-s + (−0.309 + 0.951i)7-s + (2.48 + 7.64i)8-s − 4.54·10-s + (−2.49 − 2.18i)11-s + (−1.22 + 0.888i)13-s + (−0.819 − 2.52i)14-s + (−9.11 − 6.62i)16-s + (−1.93 − 1.40i)17-s + (1.95 + 6.02i)19-s + (6.97 − 5.06i)20-s + (8.75 + 0.795i)22-s + 5.05·23-s + ⋯ |
L(s) = 1 | + (−1.51 + 1.10i)2-s + (0.777 − 2.39i)4-s + (0.619 + 0.450i)5-s + (−0.116 + 0.359i)7-s + (0.878 + 2.70i)8-s − 1.43·10-s + (−0.752 − 0.658i)11-s + (−0.339 + 0.246i)13-s + (−0.219 − 0.674i)14-s + (−2.27 − 1.65i)16-s + (−0.469 − 0.341i)17-s + (0.448 + 1.38i)19-s + (1.55 − 1.13i)20-s + (1.86 + 0.169i)22-s + 1.05·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0277048 + 0.484239i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0277048 + 0.484239i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (2.49 + 2.18i)T \) |
good | 2 | \( 1 + (2.14 - 1.55i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (-1.38 - 1.00i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (1.22 - 0.888i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.93 + 1.40i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.95 - 6.02i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 + (2.57 - 7.93i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (4.88 - 3.54i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.30 - 7.10i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.55 - 4.78i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 2.02T + 43T^{2} \) |
| 47 | \( 1 + (-1.70 - 5.25i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.04 + 3.66i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.45 - 10.6i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (11.1 + 8.11i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + (11.5 + 8.42i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.441 - 1.36i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.43 - 3.94i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.74 - 3.45i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + (9.74 - 7.08i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.56957304148033285749847602883, −9.790096196786967060379435814232, −9.082543208629966329201654054223, −8.331085875664552634015803808735, −7.43749536499936668432399915304, −6.67160343577078547522785343707, −5.82538681281014331393847203639, −5.12586466950483169183236472353, −2.90171120147881524531530674755, −1.52630953184281189018113277747,
0.41648676554171505217232410654, 1.90740630070246445279439066853, 2.77195769962929199992623669793, 4.16170152229583985852166384179, 5.47740694391306011887442459674, 7.12346215186847059051398192834, 7.54576265105133909252225010186, 8.728722856495720963202892042694, 9.349943149260382251145019524893, 9.917892359042027241805552815047