| L(s) = 1 | + (−0.528 + 1.56i)2-s + (−2.80 + 1.07i)3-s + (1.00 + 0.765i)4-s + (6.70 + 2.67i)5-s + (−0.200 − 4.95i)6-s + (6.64 − 3.07i)7-s + (−7.20 + 4.88i)8-s + (6.70 − 6.00i)9-s + (−7.73 + 9.10i)10-s + (15.4 + 4.28i)11-s + (−3.64 − 1.06i)12-s + (0.592 − 1.11i)13-s + (1.30 + 12.0i)14-s + (−21.6 − 0.299i)15-s + (−2.50 − 9.00i)16-s + (0.274 − 0.593i)17-s + ⋯ |
| L(s) = 1 | + (−0.264 + 0.783i)2-s + (−0.934 + 0.357i)3-s + (0.251 + 0.191i)4-s + (1.34 + 0.534i)5-s + (−0.0333 − 0.826i)6-s + (0.948 − 0.439i)7-s + (−0.900 + 0.610i)8-s + (0.744 − 0.667i)9-s + (−0.773 + 0.910i)10-s + (1.40 + 0.389i)11-s + (−0.303 − 0.0887i)12-s + (0.0456 − 0.0860i)13-s + (0.0934 + 0.859i)14-s + (−1.44 − 0.0199i)15-s + (−0.156 − 0.562i)16-s + (0.0161 − 0.0349i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.835002 + 1.21034i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.835002 + 1.21034i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.80 - 1.07i)T \) |
| 59 | \( 1 + (39.9 + 43.4i)T \) |
| good | 2 | \( 1 + (0.528 - 1.56i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (-6.70 - 2.67i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-6.64 + 3.07i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-15.4 - 4.28i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-0.592 + 1.11i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (-0.274 + 0.593i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (17.1 - 3.77i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (41.3 + 6.77i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (-9.51 - 28.2i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (-13.8 - 3.04i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-40.8 + 60.1i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (0.669 - 0.109i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (19.6 + 70.8i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (-24.1 + 9.64i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (42.0 - 35.7i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-77.7 - 26.2i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-23.5 - 34.7i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (50.8 - 20.2i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (20.7 - 2.25i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (99.7 - 60.0i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (35.4 - 1.92i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (10.4 + 30.8i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-84.6 - 9.20i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−12.56901620841777428310892646079, −11.65967060318198516614877108070, −10.72773334356316372546166884423, −9.848774442776488710897400772899, −8.704563964817913931566872247240, −7.24266940803565352450804016275, −6.38913820363777000175917492820, −5.70516764013149872035780191214, −4.17592922966956433853697537260, −1.88170365795955044407130025150,
1.26143169283899888069691164049, 2.07344611989520651713115550581, 4.55818099780853720072586464244, 6.05065498422219238468170593864, 6.29133070345511196609114004003, 8.280360078895376802359655589753, 9.523189155190663521708654079442, 10.19019475074899861872805907358, 11.48740153204350614126884198788, 11.74165232656930211255612462340
