L(s) = 1 | + 3.43i·2-s + (0.889 − 8.95i)3-s + 4.21·4-s − 18.6i·5-s + (30.7 + 3.05i)6-s − 68.5·7-s + 69.3i·8-s + (−79.4 − 15.9i)9-s + 63.9·10-s − 69.0i·11-s + (3.74 − 37.7i)12-s − 25.1·13-s − 235. i·14-s + (−166. − 16.5i)15-s − 170.·16-s + 489. i·17-s + ⋯ |
L(s) = 1 | + 0.858i·2-s + (0.0987 − 0.995i)3-s + 0.263·4-s − 0.744i·5-s + (0.854 + 0.0848i)6-s − 1.39·7-s + 1.08i·8-s + (−0.980 − 0.196i)9-s + 0.639·10-s − 0.570i·11-s + (0.0260 − 0.261i)12-s − 0.148·13-s − 1.20i·14-s + (−0.741 − 0.0735i)15-s − 0.667·16-s + 1.69i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0987i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1255764217\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1255764217\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.889 + 8.95i)T \) |
| 59 | \( 1 + 453. iT \) |
good | 2 | \( 1 - 3.43iT - 16T^{2} \) |
| 5 | \( 1 + 18.6iT - 625T^{2} \) |
| 7 | \( 1 + 68.5T + 2.40e3T^{2} \) |
| 11 | \( 1 + 69.0iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 25.1T + 2.85e4T^{2} \) |
| 17 | \( 1 - 489. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 352.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 736. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 166. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 776.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 432.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 3.19e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.34e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.23e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.42e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 499.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.40e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 578. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 9.31e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.81e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.62e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.93e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.69e3T + 8.85e7T^{2} \) |
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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
−12.79333973176448238506361120337, −11.76846267418433656417213845324, −10.54265893175308609379321046308, −8.983753199955976584246305914957, −8.298117513450853965094656478149, −7.15049266703071012284523187202, −6.27580064396934496504521801062, −5.57246413313920720184021496087, −3.43677680321300900689855759656, −1.81648974795155288307015657447,
0.04105365442453899755933524853, 2.60408490333450332038534657994, 3.19785706720768338397441526437, 4.54172113589759831825157677999, 6.31843254414593829314885153247, 7.11251145307497715327778732655, 9.003814914296151693245788824636, 9.909266614379147506881368051556, 10.41512927659478529306640307086, 11.33628427326135098123483626123