L(s) = 1 | + (−5.07 − 1.11i)3-s − 8·4-s − 22.0i·5-s − 34.4·7-s + (24.5 + 11.2i)9-s + (40.6 + 8.88i)12-s + (−24.4 + 111. i)15-s + 64·16-s − 61.4i·17-s + 40.2·19-s + 176. i·20-s + (174. + 38.2i)21-s − 360.·25-s + (−111. − 84.4i)27-s + 275.·28-s + 272. i·29-s + ⋯ |
L(s) = 1 | + (−0.976 − 0.213i)3-s − 4-s − 1.97i·5-s − 1.86·7-s + (0.908 + 0.417i)9-s + (0.976 + 0.213i)12-s + (−0.421 + 1.92i)15-s + 16-s − 0.876i·17-s + 0.486·19-s + 1.97i·20-s + (1.81 + 0.397i)21-s − 2.88·25-s + (−0.798 − 0.602i)27-s + 1.86·28-s + 1.74i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0207153 + 0.0166723i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0207153 + 0.0166723i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (5.07 + 1.11i)T \) |
| 59 | \( 1 - 453. iT \) |
good | 2 | \( 1 + 8T^{2} \) |
| 5 | \( 1 + 22.0iT - 125T^{2} \) |
| 7 | \( 1 + 34.4T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 61.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 40.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 272. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 458. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 632. iT - 1.48e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.18e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 + 560.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−12.64540613675810762197334675579, −11.91840140978515393363573127629, −10.13460718286402337594242598697, −9.416521741846043842526257274450, −8.703130671531721083388515498000, −7.16064898845484828680573886858, −5.69731312289455907248978754614, −5.04493462268975431818553505039, −3.83200236561842820092989515820, −0.914810407908609963011206365389,
0.01938114876531591030430636149, 3.10488000042362624741191071477, 4.03510180718320369584739869492, 5.98382194852621079607289776790, 6.40171858580779191946942758084, 7.59428558578787852750538549269, 9.583480630937734595571971488649, 9.986640743766893664643289831022, 10.81477019405885453773852343128, 11.98762802596587791184058429882