| L(s) = 1 | + (5.06 + 0.551i)2-s + (1.94 − 2.28i)3-s + (17.5 + 3.86i)4-s + (−15.9 − 12.1i)5-s + (11.1 − 10.5i)6-s + (8.66 − 21.7i)7-s + (48.2 + 16.2i)8-s + (−1.45 − 8.88i)9-s + (−74.0 − 70.1i)10-s + (13.1 − 6.10i)11-s + (42.9 − 32.6i)12-s + (−5.18 + 31.6i)13-s + (55.8 − 105. i)14-s + (−58.6 + 12.9i)15-s + (104. + 48.5i)16-s + (41.1 + 103. i)17-s + ⋯ |
| L(s) = 1 | + (1.79 + 0.194i)2-s + (0.373 − 0.440i)3-s + (2.19 + 0.483i)4-s + (−1.42 − 1.08i)5-s + (0.755 − 0.715i)6-s + (0.467 − 1.17i)7-s + (2.13 + 0.718i)8-s + (−0.0539 − 0.328i)9-s + (−2.34 − 2.21i)10-s + (0.361 − 0.167i)11-s + (1.03 − 0.785i)12-s + (−0.110 + 0.674i)13-s + (1.06 − 2.01i)14-s + (−1.00 + 0.222i)15-s + (1.64 + 0.758i)16-s + (0.587 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.14157 - 2.00649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.14157 - 2.00649i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.94 + 2.28i)T \) |
| 59 | \( 1 + (449. - 59.7i)T \) |
| good | 2 | \( 1 + (-5.06 - 0.551i)T + (7.81 + 1.71i)T^{2} \) |
| 5 | \( 1 + (15.9 + 12.1i)T + (33.4 + 120. i)T^{2} \) |
| 7 | \( 1 + (-8.66 + 21.7i)T + (-249. - 235. i)T^{2} \) |
| 11 | \( 1 + (-13.1 + 6.10i)T + (861. - 1.01e3i)T^{2} \) |
| 13 | \( 1 + (5.18 - 31.6i)T + (-2.08e3 - 701. i)T^{2} \) |
| 17 | \( 1 + (-41.1 - 103. i)T + (-3.56e3 + 3.37e3i)T^{2} \) |
| 19 | \( 1 + (-71.2 - 105. i)T + (-2.53e3 + 6.37e3i)T^{2} \) |
| 23 | \( 1 + (-7.06 + 130. i)T + (-1.20e4 - 1.31e3i)T^{2} \) |
| 29 | \( 1 + (-27.1 + 2.95i)T + (2.38e4 - 5.24e3i)T^{2} \) |
| 31 | \( 1 + (127. - 187. i)T + (-1.10e4 - 2.76e4i)T^{2} \) |
| 37 | \( 1 + (-129. + 43.4i)T + (4.03e4 - 3.06e4i)T^{2} \) |
| 41 | \( 1 + (10.0 + 185. i)T + (-6.85e4 + 7.45e3i)T^{2} \) |
| 43 | \( 1 + (414. + 191. i)T + (5.14e4 + 6.05e4i)T^{2} \) |
| 47 | \( 1 + (-334. + 254. i)T + (2.77e4 - 1.00e5i)T^{2} \) |
| 53 | \( 1 + (-45.8 + 43.4i)T + (8.06e3 - 1.48e5i)T^{2} \) |
| 61 | \( 1 + (-350. - 38.0i)T + (2.21e5 + 4.87e4i)T^{2} \) |
| 67 | \( 1 + (-643. - 216. i)T + (2.39e5 + 1.82e5i)T^{2} \) |
| 71 | \( 1 + (7.19 - 5.46i)T + (9.57e4 - 3.44e5i)T^{2} \) |
| 73 | \( 1 + (151. - 285. i)T + (-2.18e5 - 3.21e5i)T^{2} \) |
| 79 | \( 1 + (-124. - 146. i)T + (-7.97e4 + 4.86e5i)T^{2} \) |
| 83 | \( 1 + (867. - 521. i)T + (2.67e5 - 5.05e5i)T^{2} \) |
| 89 | \( 1 + (22.5 - 2.45i)T + (6.88e5 - 1.51e5i)T^{2} \) |
| 97 | \( 1 + (-75.3 - 142. i)T + (-5.12e5 + 7.55e5i)T^{2} \) |
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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
−12.29582061053465397491041325347, −11.75443918622138512758061695251, −10.60390355810049862058937678264, −8.504893562923756030748313846179, −7.68833454688556688862685383949, −6.80731449572931953206779488144, −5.28781451638942079166884598564, −4.02961906781273161814552204304, −3.75108522899620441971704787408, −1.36954725487984941167442271136,
2.72555479588551401621703315494, 3.28102169145633728639701173047, 4.58695187192849063248346043766, 5.54086785993217697860022997076, 7.03336834528310125358013644341, 7.84424118617951156350060288532, 9.549924176732109686049666181956, 11.19805812037105330619219006516, 11.47737188474919473755736269083, 12.20084047175603501420172292501
