| L(s) = 1 | + (−2.61 − 1.08i)2-s + (2.79 − 4.38i)3-s + (5.65 + 5.66i)4-s + (−1.04 − 2.86i)5-s + (−12.0 + 8.43i)6-s + (−28.5 + 5.03i)7-s + (−8.63 − 20.9i)8-s + (−11.4 − 24.4i)9-s + (−0.380 + 8.62i)10-s + (41.0 + 14.9i)11-s + (40.5 − 8.98i)12-s + (−49.8 − 41.8i)13-s + (80.0 + 17.7i)14-s + (−15.4 − 3.42i)15-s + (−0.0980 + 63.9i)16-s + (−54.6 − 31.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.923 − 0.383i)2-s + (0.536 − 0.843i)3-s + (0.706 + 0.707i)4-s + (−0.0934 − 0.256i)5-s + (−0.819 + 0.573i)6-s + (−1.54 + 0.271i)7-s + (−0.381 − 0.924i)8-s + (−0.423 − 0.905i)9-s + (−0.0120 + 0.272i)10-s + (1.12 + 0.409i)11-s + (0.976 − 0.216i)12-s + (−1.06 − 0.893i)13-s + (1.52 + 0.339i)14-s + (−0.266 − 0.0590i)15-s + (−0.00153 + 0.999i)16-s + (−0.779 − 0.449i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0660061 + 0.391482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0660061 + 0.391482i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.61 + 1.08i)T \) |
| 3 | \( 1 + (-2.79 + 4.38i)T \) |
| good | 5 | \( 1 + (1.04 + 2.86i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (28.5 - 5.03i)T + (322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-41.0 - 14.9i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (49.8 + 41.8i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (54.6 + 31.5i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (112. - 64.7i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (16.6 - 94.1i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (50.6 + 60.3i)T + (-4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-31.3 - 5.53i)T + (2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (178. - 309. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-165. + 197. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-89.4 + 245. i)T + (-6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (43.0 + 244. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 493. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-462. + 168. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (65.4 + 371. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (101. - 120. i)T + (-5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-157. + 272. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (201. + 348. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (252. + 301. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-156. + 130. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (524. - 302. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.41e3 - 514. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−12.48636519286564286215206685366, −11.91418884885921485758371993762, −10.16186719736727175381915794064, −9.332451421273338543016098709137, −8.441420305051039147887708780959, −7.10763685790794392472627471278, −6.35781121923033612853880559110, −3.56463679717399247662145279450, −2.22485164227549033477068747504, −0.25362044163853076074424624297,
2.61340464741306292945521385806, 4.24927487955474807709963690346, 6.29413253964636975458268601373, 7.10182652914583622603482777924, 8.876146149026309788495890311335, 9.264754552279938645370026117100, 10.35764392912293717120741431960, 11.22309990250060687884319711939, 12.79359593619113607094031451295, 14.25634445603451771829000433306
