| L(s) = 1 | + (1.21 − 0.410i)2-s + (0.669 + 1.59i)3-s + (−0.273 + 0.207i)4-s + (0.772 − 0.307i)5-s + (1.47 + 1.67i)6-s + (1.10 + 0.510i)7-s + (−1.69 + 2.49i)8-s + (−2.10 + 2.13i)9-s + (0.816 − 0.693i)10-s + (−1.30 − 4.68i)11-s + (−0.514 − 0.297i)12-s + (3.21 − 1.70i)13-s + (1.55 + 0.169i)14-s + (1.00 + 1.02i)15-s + (−0.854 + 3.07i)16-s + (−2.22 − 4.80i)17-s + ⋯ |
| L(s) = 1 | + (0.862 − 0.290i)2-s + (0.386 + 0.922i)3-s + (−0.136 + 0.103i)4-s + (0.345 − 0.137i)5-s + (0.601 + 0.683i)6-s + (0.416 + 0.192i)7-s + (−0.598 + 0.882i)8-s + (−0.701 + 0.712i)9-s + (0.258 − 0.219i)10-s + (−0.392 − 1.41i)11-s + (−0.148 − 0.0858i)12-s + (0.892 − 0.472i)13-s + (0.415 + 0.0452i)14-s + (0.260 + 0.265i)15-s + (−0.213 + 0.769i)16-s + (−0.539 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75671 + 0.517453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75671 + 0.517453i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.669 - 1.59i)T \) |
| 59 | \( 1 + (-7.62 - 0.964i)T \) |
| good | 2 | \( 1 + (-1.21 + 0.410i)T + (1.59 - 1.21i)T^{2} \) |
| 5 | \( 1 + (-0.772 + 0.307i)T + (3.62 - 3.43i)T^{2} \) |
| 7 | \( 1 + (-1.10 - 0.510i)T + (4.53 + 5.33i)T^{2} \) |
| 11 | \( 1 + (1.30 + 4.68i)T + (-9.42 + 5.67i)T^{2} \) |
| 13 | \( 1 + (-3.21 + 1.70i)T + (7.29 - 10.7i)T^{2} \) |
| 17 | \( 1 + (2.22 + 4.80i)T + (-11.0 + 12.9i)T^{2} \) |
| 19 | \( 1 + (-3.68 - 0.810i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (-0.277 - 1.69i)T + (-21.7 + 7.34i)T^{2} \) |
| 29 | \( 1 + (-0.917 + 2.72i)T + (-23.0 - 17.5i)T^{2} \) |
| 31 | \( 1 + (1.15 + 5.23i)T + (-28.1 + 13.0i)T^{2} \) |
| 37 | \( 1 + (6.71 - 4.55i)T + (13.6 - 34.3i)T^{2} \) |
| 41 | \( 1 + (2.78 + 0.457i)T + (38.8 + 13.0i)T^{2} \) |
| 43 | \( 1 + (-6.12 - 1.70i)T + (36.8 + 22.1i)T^{2} \) |
| 47 | \( 1 + (3.65 - 9.17i)T + (-34.1 - 32.3i)T^{2} \) |
| 53 | \( 1 + (3.72 + 3.16i)T + (8.57 + 52.3i)T^{2} \) |
| 61 | \( 1 + (-3.00 - 8.90i)T + (-48.5 + 36.9i)T^{2} \) |
| 67 | \( 1 + (-7.24 - 4.91i)T + (24.7 + 62.2i)T^{2} \) |
| 71 | \( 1 + (-2.02 - 0.805i)T + (51.5 + 48.8i)T^{2} \) |
| 73 | \( 1 + (1.25 - 11.5i)T + (-71.2 - 15.6i)T^{2} \) |
| 79 | \( 1 + (-6.41 - 3.85i)T + (37.0 + 69.7i)T^{2} \) |
| 83 | \( 1 + (-0.120 + 2.23i)T + (-82.5 - 8.97i)T^{2} \) |
| 89 | \( 1 + (5.12 + 1.72i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (0.674 + 6.20i)T + (-94.7 + 20.8i)T^{2} \) |
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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
−13.21772625279274159911700630593, −11.51560184179880485730220901511, −11.21233785251175349000290613297, −9.748943200710956340640252800848, −8.759382130200410764270976419750, −8.015097192216494952805437560293, −5.75876469517123112440133085111, −5.15105103300345524271254833048, −3.78085920632454482843468812001, −2.81210758294698042832759622785,
1.84624378087749094768536149585, 3.71928769838181407687291286664, 5.05929994068017075231362462706, 6.31095960270426619307509154952, 7.10182923445458665453893793258, 8.407902695318131881787575792116, 9.482588676561221208425604217993, 10.69547668767689151127424781904, 12.14327395998255611223166260242, 12.78270169690233909182349810450
