| L(s) = 1 | + (−0.281 + 0.204i)2-s + (−2.51 − 7.74i)3-s + (−2.43 + 7.49i)4-s + (−4.21 − 3.05i)5-s + (2.29 + 1.66i)6-s + (−2.16 + 6.65i)7-s + (−1.70 − 5.26i)8-s + (−31.8 + 23.1i)9-s + 1.81·10-s + (−15.2 + 33.1i)11-s + 64.1·12-s + (−56.9 + 41.3i)13-s + (−0.753 − 2.32i)14-s + (−13.1 + 40.3i)15-s + (−49.4 − 35.9i)16-s + (84.3 + 61.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.0996 + 0.0724i)2-s + (−0.484 − 1.49i)3-s + (−0.304 + 0.936i)4-s + (−0.376 − 0.273i)5-s + (0.156 + 0.113i)6-s + (−0.116 + 0.359i)7-s + (−0.0755 − 0.232i)8-s + (−1.18 + 0.857i)9-s + 0.0573·10-s + (−0.418 + 0.908i)11-s + 1.54·12-s + (−1.21 + 0.882i)13-s + (−0.0143 − 0.0442i)14-s + (−0.225 + 0.694i)15-s + (−0.772 − 0.561i)16-s + (1.20 + 0.874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0493665 + 0.121818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0493665 + 0.121818i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.16 - 6.65i)T \) |
| 11 | \( 1 + (15.2 - 33.1i)T \) |
| good | 2 | \( 1 + (0.281 - 0.204i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (2.51 + 7.74i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (4.21 + 3.05i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (56.9 - 41.3i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-84.3 - 61.2i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (33.0 + 101. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-25.6 + 78.9i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-125. + 90.8i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (75.8 - 233. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-15.2 - 46.8i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 30.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + (49.0 + 150. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (433. - 314. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (139. - 430. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (518. + 376. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 512.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-243. - 176. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (182. - 561. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-544. + 395. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (151. + 110. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 445.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-127. + 92.4i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−14.04739537559701677124381156888, −12.96319032750534276835083441286, −12.12264681646720165847920201402, −11.91494600963851915687424273983, −9.792407130767532903547130064610, −8.202819493039452004150068543726, −7.53650506053266469701912916830, −6.40937734537876932778896842396, −4.56609362713469519346291119489, −2.28870157101057506669731420835,
0.087062382431670002222548486597, 3.49638623640156977913199676925, 4.98958253439547118801210314351, 5.84133360509637234076042942147, 7.919403866979690672376584238049, 9.597370695403321085479392399900, 10.20994794816484516677432091880, 10.91735163825397932107053689394, 12.18190498679121374755833918835, 14.04764081875333315850991110050
