| L(s) = 1 | + (−0.816 − 1.35i)2-s + (1.06 − 2.80i)3-s + (0.699 − 1.32i)4-s + (−0.752 − 6.91i)5-s + (−4.67 + 0.844i)6-s + (−0.454 + 0.153i)7-s + (−8.68 + 0.470i)8-s + (−6.73 − 5.97i)9-s + (−8.76 + 6.66i)10-s + (9.77 + 6.62i)11-s + (−2.95 − 3.36i)12-s + (17.4 + 16.5i)13-s + (0.578 + 0.491i)14-s + (−20.2 − 5.25i)15-s + (4.37 + 6.44i)16-s + (9.50 − 28.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.678i)2-s + (0.354 − 0.934i)3-s + (0.174 − 0.330i)4-s + (−0.150 − 1.38i)5-s + (−0.778 + 0.140i)6-s + (−0.0648 + 0.0218i)7-s + (−1.08 + 0.0588i)8-s + (−0.748 − 0.663i)9-s + (−0.876 + 0.666i)10-s + (0.888 + 0.602i)11-s + (−0.246 − 0.280i)12-s + (1.34 + 1.27i)13-s + (0.0412 + 0.0350i)14-s + (−1.34 − 0.350i)15-s + (0.273 + 0.403i)16-s + (0.558 − 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00484i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00324217 - 1.33901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00324217 - 1.33901i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.06 + 2.80i)T \) |
| 59 | \( 1 + (-45.3 + 37.7i)T \) |
| good | 2 | \( 1 + (0.816 + 1.35i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (0.752 + 6.91i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (0.454 - 0.153i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (-9.77 - 6.62i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (-17.4 - 16.5i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (-9.50 + 28.1i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (-3.61 - 22.0i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (15.4 - 4.28i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (4.85 - 8.06i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (-5.02 + 30.6i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-2.48 + 45.8i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (-14.8 - 4.12i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (19.9 + 29.4i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (0.359 - 3.30i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (22.1 - 29.1i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (-69.2 + 41.6i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (-5.29 - 97.7i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-3.96 + 36.4i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (-67.1 + 79.0i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (26.8 - 67.3i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (-28.7 - 62.1i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (24.0 - 39.9i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (6.99 + 8.24i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−11.92156864885749930806336254761, −11.39229544447164886165012541824, −9.514318002148008863735468366508, −9.199417665117463047452232462616, −8.099973002429546749645826271285, −6.72238601995374299854201195180, −5.61542661899501658742825202662, −3.85533847693570695212480037101, −1.89068403560041722673586978667, −0.954171162755186106267623124381,
3.09419006944024512945286668657, 3.66485850817565347627895632623, 5.88698101314706214166206187376, 6.64968197149515058899757595076, 8.086442464027995441857939695061, 8.589752163445989515496528100379, 9.991116062735032882922111825921, 10.85874250871643707965763635082, 11.57429639157785896026803333380, 13.13457074470733347332512790938
