| L(s) = 1 | + (0.658 + 0.759i)3-s + (−1.68 − 11.7i)5-s + (−15.1 + 33.0i)7-s + (3.69 − 25.7i)9-s + (68.8 + 20.2i)11-s + (8.11 + 17.7i)13-s + (7.80 − 9.00i)15-s + (59.0 + 37.9i)17-s + (54.4 − 35.0i)19-s + (−35.0 + 10.2i)21-s + (25.1 − 107. i)23-s + (−14.9 + 4.38i)25-s + (44.8 − 28.7i)27-s + (115. + 74.0i)29-s + (−127. + 146. i)31-s + ⋯ |
| L(s) = 1 | + (0.126 + 0.146i)3-s + (−0.150 − 1.04i)5-s + (−0.815 + 1.78i)7-s + (0.136 − 0.952i)9-s + (1.88 + 0.553i)11-s + (0.173 + 0.379i)13-s + (0.134 − 0.155i)15-s + (0.842 + 0.541i)17-s + (0.657 − 0.422i)19-s + (−0.364 + 0.107i)21-s + (0.228 − 0.973i)23-s + (−0.119 + 0.0350i)25-s + (0.319 − 0.205i)27-s + (0.738 + 0.474i)29-s + (−0.736 + 0.850i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.78718 + 0.286304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78718 + 0.286304i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-25.1 + 107. i)T \) |
| good | 3 | \( 1 + (-0.658 - 0.759i)T + (-3.84 + 26.7i)T^{2} \) |
| 5 | \( 1 + (1.68 + 11.7i)T + (-119. + 35.2i)T^{2} \) |
| 7 | \( 1 + (15.1 - 33.0i)T + (-224. - 259. i)T^{2} \) |
| 11 | \( 1 + (-68.8 - 20.2i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (-8.11 - 17.7i)T + (-1.43e3 + 1.66e3i)T^{2} \) |
| 17 | \( 1 + (-59.0 - 37.9i)T + (2.04e3 + 4.46e3i)T^{2} \) |
| 19 | \( 1 + (-54.4 + 35.0i)T + (2.84e3 - 6.23e3i)T^{2} \) |
| 29 | \( 1 + (-115. - 74.0i)T + (1.01e4 + 2.21e4i)T^{2} \) |
| 31 | \( 1 + (127. - 146. i)T + (-4.23e3 - 2.94e4i)T^{2} \) |
| 37 | \( 1 + (14.5 - 101. i)T + (-4.86e4 - 1.42e4i)T^{2} \) |
| 41 | \( 1 + (-17.3 - 120. i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-207. - 239. i)T + (-1.13e4 + 7.86e4i)T^{2} \) |
| 47 | \( 1 - 367.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (22.1 - 48.5i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (193. + 424. i)T + (-1.34e5 + 1.55e5i)T^{2} \) |
| 61 | \( 1 + (-89.7 + 103. i)T + (-3.23e4 - 2.24e5i)T^{2} \) |
| 67 | \( 1 + (804. - 236. i)T + (2.53e5 - 1.62e5i)T^{2} \) |
| 71 | \( 1 + (877. - 257. i)T + (3.01e5 - 1.93e5i)T^{2} \) |
| 73 | \( 1 + (-176. + 113. i)T + (1.61e5 - 3.53e5i)T^{2} \) |
| 79 | \( 1 + (-115. - 253. i)T + (-3.22e5 + 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-10.6 + 73.9i)T + (-5.48e5 - 1.61e5i)T^{2} \) |
| 89 | \( 1 + (247. + 285. i)T + (-1.00e5 + 6.97e5i)T^{2} \) |
| 97 | \( 1 + (64.5 + 448. i)T + (-8.75e5 + 2.57e5i)T^{2} \) |
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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
−12.36819873899613895851511647274, −11.69549656427632867247783668422, −9.781759369813383906410991446870, −8.995150158362508573738519056159, −8.758010411893994219605063249238, −6.75984007311175922813995106886, −5.91732716167254550405794186371, −4.53116375039016391348370904406, −3.20085982920282955537683831772, −1.27749428291286655132308018920,
1.03612149188255839064522876802, 3.23991246648789287684836630988, 4.01535153725161692540893571740, 5.98684870041306136718289455596, 7.18434721666543385182704190301, 7.52440157575083402126532219080, 9.288872932140711939043762545399, 10.29707705362533291233980885416, 10.94628672552554061478426182753, 11.98293119854367684357932777834
