L(s) = 1 | − 3-s + 5-s + (0.827 + 1.43i)7-s + 9-s + (−2.81 − 4.87i)11-s + (0.800 − 1.38i)13-s − 15-s + (−1.35 + 2.34i)17-s + (−0.454 + 0.787i)19-s + (−0.827 − 1.43i)21-s + (−3.02 + 5.23i)23-s + 25-s − 27-s + (−1.34 − 2.33i)29-s + (5.11 + 8.85i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + (0.312 + 0.541i)7-s + 0.333·9-s + (−0.848 − 1.46i)11-s + (0.222 − 0.384i)13-s − 0.258·15-s + (−0.328 + 0.569i)17-s + (−0.104 + 0.180i)19-s + (−0.180 − 0.312i)21-s + (−0.630 + 1.09i)23-s + 0.200·25-s − 0.192·27-s + (−0.250 − 0.433i)29-s + (0.918 + 1.59i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.471914666\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.471914666\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-8.03 + 1.56i)T \) |
good | 7 | \( 1 + (-0.827 - 1.43i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.81 + 4.87i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.800 + 1.38i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.35 - 2.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.454 - 0.787i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.02 - 5.23i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.34 + 2.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.11 - 8.85i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.237 - 0.411i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.573 - 0.992i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 5.46T + 43T^{2} \) |
| 47 | \( 1 + (-1.69 - 2.94i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 5.97T + 53T^{2} \) |
| 59 | \( 1 - 8.25T + 59T^{2} \) |
| 61 | \( 1 + (-5.07 + 8.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-1.81 - 3.14i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.751 - 1.30i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.38 - 7.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.96 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + (-2.33 + 4.03i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.245870066058211816054628339675, −8.129177275889083165596531638794, −6.89051105640831595605481999189, −6.10084234186512275327681164028, −5.57951792316504460638792741777, −5.08260823709505291736694019900, −3.89907380302460700497812162886, −3.02396411801247496114293711656, −2.03074699490986513008911496682, −0.869432823657595475603136525370,
0.58635853988780008527197485354, 1.93817508858286494727572929380, 2.58309619486762282626130739401, 4.14536025651149287436368292552, 4.54999930040703704331165008793, 5.34177557544895644643484831911, 6.19098278992304165711391627146, 6.94049506044008897202811779871, 7.50152284210648014166462221326, 8.257229603465600220375659625298