L(s) = 1 | + (−1.40 + 0.144i)2-s − i·3-s + (1.95 − 0.406i)4-s + (0.144 + 1.40i)6-s + 3.62·7-s + (−2.69 + 0.855i)8-s − 9-s − 6.20i·11-s + (−0.406 − 1.95i)12-s − 0.578i·13-s + (−5.10 + 0.524i)14-s + (3.66 − 1.59i)16-s − 1.42·17-s + (1.40 − 0.144i)18-s + 5.62i·19-s + ⋯ |
L(s) = 1 | + (−0.994 + 0.102i)2-s − 0.577i·3-s + (0.979 − 0.203i)4-s + (0.0590 + 0.574i)6-s + 1.37·7-s + (−0.953 + 0.302i)8-s − 0.333·9-s − 1.87i·11-s + (−0.117 − 0.565i)12-s − 0.160i·13-s + (−1.36 + 0.140i)14-s + (0.917 − 0.398i)16-s − 0.344·17-s + (0.331 − 0.0340i)18-s + 1.29i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.844521 - 0.618052i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.844521 - 0.618052i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 0.144i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 + 6.20iT - 11T^{2} \) |
| 13 | \( 1 + 0.578iT - 13T^{2} \) |
| 17 | \( 1 + 1.42T + 17T^{2} \) |
| 19 | \( 1 - 5.62iT - 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 + 7.83iT - 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 + 7.25iT - 43T^{2} \) |
| 47 | \( 1 + 6.78T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 2.20iT - 59T^{2} \) |
| 61 | \( 1 + 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 - 3.25iT - 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
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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
−10.89861549090212939331677662967, −9.402815800720190397725683938410, −8.417776423194298973240818501908, −8.140394407826987335262403315633, −7.17065160887224147619728458966, −6.04391569230839519112243752601, −5.32915562520870735439483302970, −3.48783034461214496032036352762, −2.08732678671105850208355199259, −0.873423304839546583199217713719,
1.55453057656037348169259121235, 2.70964227011762655971627405324, 4.44402680576877082698655176151, 5.09491099169146447756957214220, 6.71501099131561653085446243717, 7.42025464417714180457156283080, 8.340686412981253728349330453208, 9.216001210864604849294216656828, 9.827301003361831681847558985509, 10.89676938187121560673782679705