| L(s) = 1 | + (−0.982 + 0.357i)2-s + (−5.19 + 0.0395i)3-s + (−5.29 + 4.43i)4-s + (2.68 + 15.1i)5-s + (5.09 − 1.89i)6-s + (−18.0 − 15.1i)7-s + (7.79 − 13.5i)8-s + (26.9 − 0.411i)9-s + (−8.07 − 13.9i)10-s + (−7.09 + 40.2i)11-s + (27.3 − 23.2i)12-s + (11.9 + 4.34i)13-s + (23.1 + 8.42i)14-s + (−14.5 − 78.8i)15-s + (6.76 − 38.3i)16-s + (23.8 + 41.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.347 + 0.126i)2-s + (−0.999 + 0.00761i)3-s + (−0.661 + 0.554i)4-s + (0.239 + 1.35i)5-s + (0.346 − 0.129i)6-s + (−0.973 − 0.817i)7-s + (0.344 − 0.596i)8-s + (0.999 − 0.0152i)9-s + (−0.255 − 0.442i)10-s + (−0.194 + 1.10i)11-s + (0.657 − 0.559i)12-s + (0.254 + 0.0926i)13-s + (0.441 + 0.160i)14-s + (−0.250 − 1.35i)15-s + (0.105 − 0.599i)16-s + (0.339 + 0.588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.161886 + 0.421456i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.161886 + 0.421456i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.19 - 0.0395i)T \) |
| good | 2 | \( 1 + (0.982 - 0.357i)T + (6.12 - 5.14i)T^{2} \) |
| 5 | \( 1 + (-2.68 - 15.1i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (18.0 + 15.1i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (7.09 - 40.2i)T + (-1.25e3 - 455. i)T^{2} \) |
| 13 | \( 1 + (-11.9 - 4.34i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-23.8 - 41.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (67.6 - 117. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-36.2 + 30.4i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (118. - 43.2i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-158. + 132. i)T + (5.17e3 - 2.93e4i)T^{2} \) |
| 37 | \( 1 + (-93.6 - 162. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (41.7 + 15.2i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-41.2 + 234. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-83.8 - 70.3i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 - 11.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-126. - 715. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (45.5 + 38.2i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (290. + 105. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-130. - 226. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-186. + 323. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (128. - 46.9i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-1.03e3 + 376. i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (716. - 1.24e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-137. + 780. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−17.27801003991533407114968610624, −16.55811163766091200376595077354, −14.98077671950524233024347306635, −13.40431806466835301868225041324, −12.35447376669603795956783313097, −10.49264699395342300347403903349, −9.918720550596395109061919555930, −7.46951790572356534145568221439, −6.40305121736954872508937963713, −3.94417962578661271917255253958,
0.57748247998975939620741274177, 4.96382607162700601833198861256, 6.01332145356592420027285843104, 8.729086011840217425227170297172, 9.610600992045367482335563499888, 11.17044326003117114345156221392, 12.69031803200426040689283584116, 13.44163055549524274831034739254, 15.61381981849628426443429281689, 16.52826453000411116669349780229
