| L(s) = 1 | + (−1.30 − 0.951i)2-s + (−0.913 − 0.406i)3-s + (0.190 + 0.587i)4-s + (0.190 − 0.330i)5-s + (0.809 + 1.40i)6-s + (2.00 − 2.22i)7-s + (−0.690 + 2.12i)8-s + (−1.33 − 1.48i)9-s + (−0.564 + 0.251i)10-s + (5.12 + 1.08i)11-s + (0.0646 − 0.614i)12-s + (0.193 + 1.84i)13-s + (−4.74 + 1.00i)14-s + (−0.309 + 0.224i)15-s + (3.92 − 2.85i)16-s + (4.14 − 0.880i)17-s + ⋯ |
| L(s) = 1 | + (−0.925 − 0.672i)2-s + (−0.527 − 0.234i)3-s + (0.0954 + 0.293i)4-s + (0.0854 − 0.147i)5-s + (0.330 + 0.572i)6-s + (0.758 − 0.842i)7-s + (−0.244 + 0.751i)8-s + (−0.446 − 0.495i)9-s + (−0.178 + 0.0794i)10-s + (1.54 + 0.328i)11-s + (0.0186 − 0.177i)12-s + (0.0537 + 0.511i)13-s + (−1.26 + 0.269i)14-s + (−0.0797 + 0.0579i)15-s + (0.981 − 0.713i)16-s + (1.00 − 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.705365 - 0.624314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.705365 - 0.624314i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (1.30 + 0.951i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.913 + 0.406i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (-0.190 + 0.330i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.00 + 2.22i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (-5.12 - 1.08i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.193 - 1.84i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-4.14 + 0.880i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.522 - 4.97i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.07 + 3.30i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.16 - 3.75i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-2.11 - 3.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.25 - 1.00i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.248 + 2.36i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-4.54 + 3.30i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.473 - 0.526i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-0.482 - 0.214i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + (0.118 - 0.204i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.42 + 8.24i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (11.3 + 2.40i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-6.47 + 2.88i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (2.66 + 8.19i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (5.78 + 17.7i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.941881202825190544521570228253, −9.037132872727097052558686685198, −8.488271179700213122009746456255, −7.38639503951129002323297707417, −6.52289595412904176070809739905, −5.54347109230729054470846047083, −4.47045329453765318506814609737, −3.30815963737098415541682036502, −1.56426143540442415862744340013, −0.995672606876184435381753046402,
0.983601736695836751077992218048, 2.72329667748480717692769130805, 4.09782730757576199477479132867, 5.28852988034680229549321081070, 6.04836945944254229936874529379, 6.84524431927002063341350690719, 7.934129081052755818457352847821, 8.520623134688514438123398002398, 9.171659774267823531273155483430, 10.04366007873856892863696227115
