| L(s) = 1 | + (−2.02 − 2.98i)2-s + (0.162 − 2.99i)3-s + (−1.86 + 4.68i)4-s + (−15.6 + 7.25i)5-s + (−9.28 + 5.58i)6-s + (−2.35 + 8.46i)7-s + (−10.4 + 2.29i)8-s + (−8.94 − 0.973i)9-s + (53.5 + 32.1i)10-s + (23.4 − 22.2i)11-s + (13.7 + 6.35i)12-s + (27.9 − 3.04i)13-s + (30.0 − 10.1i)14-s + (19.1 + 48.1i)15-s + (57.2 + 54.2i)16-s + (4.60 + 16.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.716 − 1.05i)2-s + (0.0312 − 0.576i)3-s + (−0.233 + 0.585i)4-s + (−1.40 + 0.649i)5-s + (−0.631 + 0.380i)6-s + (−0.126 + 0.457i)7-s + (−0.460 + 0.101i)8-s + (−0.331 − 0.0360i)9-s + (1.69 + 1.01i)10-s + (0.643 − 0.609i)11-s + (0.330 + 0.152i)12-s + (0.597 − 0.0649i)13-s + (0.574 − 0.193i)14-s + (0.330 + 0.829i)15-s + (0.895 + 0.847i)16-s + (0.0656 + 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.650646 - 0.130504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.650646 - 0.130504i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.162 + 2.99i)T \) |
| 59 | \( 1 + (-15.9 - 452. i)T \) |
| good | 2 | \( 1 + (2.02 + 2.98i)T + (-2.96 + 7.43i)T^{2} \) |
| 5 | \( 1 + (15.6 - 7.25i)T + (80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (2.35 - 8.46i)T + (-293. - 176. i)T^{2} \) |
| 11 | \( 1 + (-23.4 + 22.2i)T + (72.0 - 1.32e3i)T^{2} \) |
| 13 | \( 1 + (-27.9 + 3.04i)T + (2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (-4.60 - 16.5i)T + (-4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (-4.49 + 3.41i)T + (1.83e3 - 6.60e3i)T^{2} \) |
| 23 | \( 1 + (-13.1 + 24.7i)T + (-6.82e3 - 1.00e4i)T^{2} \) |
| 29 | \( 1 + (129. - 190. i)T + (-9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (-176. - 134. i)T + (7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (-193. - 42.5i)T + (4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (188. + 354. i)T + (-3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (7.88 + 7.46i)T + (4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (-457. - 211. i)T + (6.72e4 + 7.91e4i)T^{2} \) |
| 53 | \( 1 + (300. - 181. i)T + (6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (-92.1 - 135. i)T + (-8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (361. - 79.5i)T + (2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (-510. - 236. i)T + (2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (829. - 279. i)T + (3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (-43.1 - 795. i)T + (-4.90e5 + 5.33e4i)T^{2} \) |
| 83 | \( 1 + (-169. - 1.03e3i)T + (-5.41e5 + 1.82e5i)T^{2} \) |
| 89 | \( 1 + (-324. + 477. i)T + (-2.60e5 - 6.54e5i)T^{2} \) |
| 97 | \( 1 + (-146. - 49.4i)T + (7.26e5 + 5.52e5i)T^{2} \) |
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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
−11.93250405836827453146236371014, −11.20474720949155867066702088856, −10.54506853675320004600612239494, −9.010162212095674397056631573671, −8.378927607934273435821531226819, −7.16877558515447080789928670279, −5.97760697536118021918552224170, −3.77238464441958443329290046002, −2.80754286707826892973163292988, −1.04387430517138566867328489161,
0.49418784358390207857071083684, 3.61701926129891473108419089020, 4.58316373468100415044733345859, 6.18070584568491602566846959510, 7.41291095348937519301369126208, 8.093203829723154140255577341692, 9.037794713252503340318418337509, 9.899795234712786756656705475037, 11.41134753498114517996490721276, 12.03677430500474733078517884885
