| L(s) = 1 | + (−1.54 − 0.893i)2-s + (5.18 − 0.358i)3-s + (−2.40 − 4.16i)4-s − 16.9·5-s + (−8.34 − 4.07i)6-s + (−9.83 − 15.6i)7-s + 22.8i·8-s + (26.7 − 3.72i)9-s + (26.2 + 15.1i)10-s − 7.14i·11-s + (−13.9 − 20.7i)12-s + (−67.0 − 38.6i)13-s + (1.19 + 33.0i)14-s + (−87.8 + 6.08i)15-s + (1.24 − 2.16i)16-s + (1.06 − 1.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.547 − 0.316i)2-s + (0.997 − 0.0690i)3-s + (−0.300 − 0.520i)4-s − 1.51·5-s + (−0.567 − 0.277i)6-s + (−0.531 − 0.847i)7-s + 1.01i·8-s + (0.990 − 0.137i)9-s + (0.829 + 0.478i)10-s − 0.195i·11-s + (−0.335 − 0.498i)12-s + (−1.42 − 0.825i)13-s + (0.0229 + 0.631i)14-s + (−1.51 + 0.104i)15-s + (0.0195 − 0.0337i)16-s + (0.0151 − 0.0263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.161031 - 0.664079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.161031 - 0.664079i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.18 + 0.358i)T \) |
| 7 | \( 1 + (9.83 + 15.6i)T \) |
| good | 2 | \( 1 + (1.54 + 0.893i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + 16.9T + 125T^{2} \) |
| 11 | \( 1 + 7.14iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (67.0 + 38.6i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-1.06 + 1.84i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-104. + 60.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 10.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (110. - 63.9i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-184. + 106. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (12.2 + 21.1i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (64.8 - 112. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-86.4 - 149. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-202. + 350. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (291. + 168. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (330. + 572. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (300. + 173. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-134. - 233. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 637. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (503. + 290. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-85.9 + 148. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (635. + 1.10e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (175. + 303. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (73.3 - 42.3i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.12691000281147602020241295370, −12.94289363277072844816338336238, −11.60131965915164567904038488041, −10.29356674817087571026413280277, −9.392996567077980970162481695277, −8.026518676191172850793851217554, −7.28278031667303064780101415439, −4.65905867498632408316178187488, −3.13041056498669651177916431141, −0.50730426078986587103904544884,
3.09882864498379871754827044259, 4.37483315767730306969582834819, 7.18077907017240160691931031046, 7.84529528284244748042696085139, 8.970620681322347003864207010317, 9.802816511242224347748283405064, 11.99209659219827777093631307867, 12.44223008636439867967353797187, 13.99522634006980244653308434023, 15.25930025821016337210904586207
