| L(s) = 1 | + (2.26 + 1.89i)2-s + (0.124 + 0.707i)4-s + (0.0925 − 0.524i)5-s + (−11.8 − 20.6i)7-s + (10.7 − 18.6i)8-s + (1.20 − 1.01i)10-s + (18.5 − 32.1i)11-s + (−14.9 − 5.45i)13-s + (12.1 − 69.1i)14-s + (65.0 − 23.6i)16-s + (47.7 + 40.0i)17-s + (−57.8 − 59.2i)19-s + 0.383·20-s + (103. − 37.5i)22-s + (3.18 + 18.0i)23-s + ⋯ |
| L(s) = 1 | + (0.799 + 0.671i)2-s + (0.0155 + 0.0884i)4-s + (0.00827 − 0.0469i)5-s + (−0.642 − 1.11i)7-s + (0.475 − 0.822i)8-s + (0.0381 − 0.0319i)10-s + (0.509 − 0.882i)11-s + (−0.319 − 0.116i)13-s + (0.232 − 1.32i)14-s + (1.01 − 0.369i)16-s + (0.680 + 0.571i)17-s + (−0.698 − 0.715i)19-s + 0.00428·20-s + (0.999 − 0.363i)22-s + (0.0288 + 0.163i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.09146 - 0.861839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09146 - 0.861839i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (57.8 + 59.2i)T \) |
| good | 2 | \( 1 + (-2.26 - 1.89i)T + (1.38 + 7.87i)T^{2} \) |
| 5 | \( 1 + (-0.0925 + 0.524i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (11.8 + 20.6i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-18.5 + 32.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (14.9 + 5.45i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-47.7 - 40.0i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-3.18 - 18.0i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (13.6 - 11.4i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (76.5 + 132. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 41.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-314. + 114. i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (53.2 - 302. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (37.2 - 31.2i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-84.6 - 479. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (160. + 134. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (36.3 + 206. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-327. + 274. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (105. - 599. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (299. - 108. i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (245. - 89.2i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (328. + 569. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-1.24e3 - 453. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-245. - 206. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−12.66488324610759191096042906827, −11.09587611205882123285931376844, −10.25016411005684070991410667815, −9.160863532128846504032005951764, −7.64311048969813037524276206468, −6.69424263833289334525464162356, −5.81572040124483874716985266719, −4.45185940846707900770729038209, −3.43218196384240578134670989839, −0.841231642168258121166503088372,
2.08756055570147832471693316551, 3.23405463359844097098062038806, 4.56422459617708560126356687424, 5.70909106780394235957308705266, 7.06166264615123890038693600795, 8.456674514796951504943171912483, 9.504883845417122866334133188872, 10.60552497933327864500231719781, 11.93405623042659393927769488234, 12.31009507580737941762631129222
