L(s) = 1 | + (1.84 + 0.786i)2-s + (1.38 + 1.45i)4-s + (−4.22 − 1.37i)5-s + (−0.831 − 0.951i)7-s + (0.00608 + 0.0162i)8-s + (−6.69 − 5.85i)10-s + (−3.87 + 1.07i)11-s + (0.950 − 3.44i)13-s + (−0.781 − 2.40i)14-s + (0.178 − 3.97i)16-s + (3.04 − 2.21i)17-s + (−0.376 − 0.699i)19-s + (−3.87 − 8.03i)20-s + (−7.98 − 1.08i)22-s + (0.166 − 0.730i)23-s + ⋯ |
L(s) = 1 | + (1.30 + 0.556i)2-s + (0.694 + 0.725i)4-s + (−1.88 − 0.613i)5-s + (−0.314 − 0.359i)7-s + (0.00215 + 0.00573i)8-s + (−2.11 − 1.85i)10-s + (−1.16 + 0.322i)11-s + (0.263 − 0.955i)13-s + (−0.208 − 0.643i)14-s + (0.0446 − 0.994i)16-s + (0.738 − 0.536i)17-s + (−0.0863 − 0.160i)19-s + (−0.865 − 1.79i)20-s + (−1.70 − 0.230i)22-s + (0.0347 − 0.152i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.635094 - 0.831157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.635094 - 0.831157i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 71 | \( 1 + (0.314 - 8.42i)T \) |
good | 2 | \( 1 + (-1.84 - 0.786i)T + (1.38 + 1.44i)T^{2} \) |
| 5 | \( 1 + (4.22 + 1.37i)T + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (0.831 + 0.951i)T + (-0.939 + 6.93i)T^{2} \) |
| 11 | \( 1 + (3.87 - 1.07i)T + (9.44 - 5.64i)T^{2} \) |
| 13 | \( 1 + (-0.950 + 3.44i)T + (-11.1 - 6.66i)T^{2} \) |
| 17 | \( 1 + (-3.04 + 2.21i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.376 + 0.699i)T + (-10.4 + 15.8i)T^{2} \) |
| 23 | \( 1 + (-0.166 + 0.730i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (8.66 - 1.17i)T + (27.9 - 7.71i)T^{2} \) |
| 31 | \( 1 + (7.23 - 0.324i)T + (30.8 - 2.77i)T^{2} \) |
| 37 | \( 1 + (-1.78 - 7.82i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-1.65 - 2.07i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (5.27 + 0.474i)T + (42.3 + 7.67i)T^{2} \) |
| 47 | \( 1 + (2.61 - 0.474i)T + (44.0 - 16.5i)T^{2} \) |
| 53 | \( 1 + (-1.41 + 1.48i)T + (-2.37 - 52.9i)T^{2} \) |
| 59 | \( 1 + (3.56 + 5.39i)T + (-23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (-2.09 + 2.40i)T + (-8.18 - 60.4i)T^{2} \) |
| 67 | \( 1 + (-8.94 + 8.55i)T + (3.00 - 66.9i)T^{2} \) |
| 73 | \( 1 + (-2.44 + 5.71i)T + (-50.4 - 52.7i)T^{2} \) |
| 79 | \( 1 + (-10.5 + 3.96i)T + (59.4 - 51.9i)T^{2} \) |
| 83 | \( 1 + (-10.3 + 6.84i)T + (32.6 - 76.3i)T^{2} \) |
| 89 | \( 1 + (-2.40 - 2.30i)T + (3.99 + 88.9i)T^{2} \) |
| 97 | \( 1 + (-12.4 - 9.89i)T + (21.5 + 94.5i)T^{2} \) |
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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
−10.58393961590013130796131706766, −9.378983222120380611390865298854, −8.007239935386074185598013837362, −7.67612910181222567020738095206, −6.79334792610391525955267257885, −5.32246433103921265104836486071, −4.91109123681947770952306102385, −3.72517829822706075720888263949, −3.23078931303766719347086007565, −0.35964300465874634694461288301,
2.39925172854166871858948511069, 3.57433521996312652364834959851, 3.88614117684064301235102401286, 5.11805546945772068662791040325, 6.08854881989656357488095137316, 7.32499599391478643739685849188, 7.998169523943052871636782188317, 9.010313708676123104006444004987, 10.55287112488171676637498821499, 11.14320448696962665848987342970