L(s) = 1 | + (0.755 + 0.654i)2-s + (0.989 − 0.142i)3-s + (0.142 + 0.989i)4-s + (−0.281 + 0.959i)5-s + (0.841 + 0.540i)6-s + (−0.540 + 0.841i)8-s + (0.959 − 0.281i)9-s + (−0.841 + 0.540i)10-s + (0.281 + 0.959i)12-s + (−0.142 + 0.989i)15-s + (−0.959 + 0.281i)16-s + (−0.755 − 0.345i)17-s + (0.909 + 0.415i)18-s + (−0.544 − 1.19i)19-s + (−0.989 − 0.142i)20-s + ⋯ |
L(s) = 1 | + (0.755 + 0.654i)2-s + (0.989 − 0.142i)3-s + (0.142 + 0.989i)4-s + (−0.281 + 0.959i)5-s + (0.841 + 0.540i)6-s + (−0.540 + 0.841i)8-s + (0.959 − 0.281i)9-s + (−0.841 + 0.540i)10-s + (0.281 + 0.959i)12-s + (−0.142 + 0.989i)15-s + (−0.959 + 0.281i)16-s + (−0.755 − 0.345i)17-s + (0.909 + 0.415i)18-s + (−0.544 − 1.19i)19-s + (−0.989 − 0.142i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.059886907\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.059886907\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.755 - 0.654i)T \) |
| 3 | \( 1 + (-0.989 + 0.142i)T \) |
| 5 | \( 1 + (0.281 - 0.959i)T \) |
| 23 | \( 1 + (0.540 + 0.841i)T \) |
good | 7 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 11 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.755 + 0.345i)T + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (-1.49 - 0.215i)T + (0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 - 1.68iT - T^{2} \) |
| 53 | \( 1 + (-1.27 - 1.10i)T + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (-1.07 - 0.153i)T + (0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (1.25 + 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (1.89 - 0.557i)T + (0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (0.841 + 0.540i)T^{2} \) |
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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
−9.879774148544559560266427649542, −8.800635539812696745081056966157, −8.285449986039602807744766878481, −7.33825426232417507381955075818, −6.81585631408945222558681284307, −6.12921653190200053962411457584, −4.63938154721005546425678867659, −4.07179316521485776434082872496, −2.88800159536392013868473406188, −2.42417610226503866323467457320,
1.46371333911714546572000144717, 2.40932342191619966262381874123, 3.72948311000713283926289227190, 4.16220166879366969082371544873, 5.11055797280014996622541608332, 6.08095230423935097949998535520, 7.17800813123031985411720819630, 8.286442945030976019392086338227, 8.712766281998653418440993536336, 9.827859461109248523859724496743