L(s) = 1 | + (−0.324 − 0.235i)2-s + (−0.568 − 1.74i)4-s + (−0.0534 − 2.23i)5-s − 3.13·7-s + (−0.476 + 1.46i)8-s + (−0.509 + 0.738i)10-s + (0.141 + 0.103i)11-s + (−1.01 + 0.738i)13-s + (1.01 + 0.738i)14-s + (−2.47 + 1.79i)16-s + (2.01 − 6.19i)17-s + (−1.51 + 4.67i)19-s + (−3.87 + 1.36i)20-s + (−0.0217 − 0.0669i)22-s + (−2.48 − 1.80i)23-s + ⋯ |
L(s) = 1 | + (−0.229 − 0.166i)2-s + (−0.284 − 0.874i)4-s + (−0.0239 − 0.999i)5-s − 1.18·7-s + (−0.168 + 0.517i)8-s + (−0.161 + 0.233i)10-s + (0.0427 + 0.0310i)11-s + (−0.281 + 0.204i)13-s + (0.271 + 0.197i)14-s + (−0.618 + 0.449i)16-s + (0.488 − 1.50i)17-s + (−0.348 + 1.07i)19-s + (−0.867 + 0.304i)20-s + (−0.00463 − 0.0142i)22-s + (−0.517 − 0.375i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0908472 + 0.246513i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0908472 + 0.246513i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.0534 + 2.23i)T \) |
good | 2 | \( 1 + (0.324 + 0.235i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 + (-0.141 - 0.103i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.01 - 0.738i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.01 + 6.19i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.51 - 4.67i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.48 + 1.80i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.10 - 9.54i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.54 - 4.74i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.72 + 1.25i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.44 + 1.77i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 6.72T + 43T^{2} \) |
| 47 | \( 1 + (3.99 + 12.3i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.37 - 4.23i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (10.5 - 7.68i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.64 + 3.37i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (2.19 - 6.75i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.96 + 6.04i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.56 + 2.59i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (3.71 + 11.4i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.31 + 7.12i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (6.44 + 4.68i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.86 + 14.9i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.948269056060291261826730636596, −9.168849158721215033477229969105, −8.616412602336224674786696780716, −7.29886084403266251577707454221, −6.26280788777141442643068092393, −5.37611108844132727442516438762, −4.56573413467339294396568953625, −3.18909424656994113450577478359, −1.58998017556422243179795670104, −0.14842449454794481469159626332,
2.56530373104149043263546453186, 3.42496626742184322754505450224, 4.29599767938791146378051608128, 6.08857972589106406487816419534, 6.56273504942986539186077453996, 7.63738015361296559284958354317, 8.230516730709372722450457365746, 9.523448393478759843994873371297, 9.900879698205694657502035405409, 10.98227794780825304216548173734