| 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
−11.98293119854367684357932777834, −10.94628672552554061478426182753, −10.29707705362533291233980885416, −9.288872932140711939043762545399, −7.52440157575083402126532219080, −7.18434721666543385182704190301, −5.98684870041306136718289455596, −4.01535153725161692540893571740, −3.23991246648789287684836630988, −1.03612149188255839064522876802,
1.27749428291286655132308018920, 3.20085982920282955537683831772, 4.53116375039016391348370904406, 5.91732716167254550405794186371, 6.75984007311175922813995106886, 8.758010411893994219605063249238, 8.995150158362508573738519056159, 9.781759369813383906410991446870, 11.69549656427632867247783668422, 12.36819873899613895851511647274
