L(s) = 1 | + (−0.995 + 0.0950i)2-s + (1.65 − 1.06i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−1.54 + 1.21i)6-s + (−1.03 − 1.44i)7-s + (−0.959 + 0.281i)8-s + (1.18 − 2.59i)9-s + (0.0475 − 0.998i)10-s + (1.42 − 1.35i)12-s + (1.16 + 1.34i)14-s + (0.815 + 1.78i)15-s + (0.928 − 0.371i)16-s + (−0.934 + 2.69i)18-s + (0.0475 + 0.998i)20-s + (−3.24 − 1.29i)21-s + ⋯ |
L(s) = 1 | + (−0.995 + 0.0950i)2-s + (1.65 − 1.06i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−1.54 + 1.21i)6-s + (−1.03 − 1.44i)7-s + (−0.959 + 0.281i)8-s + (1.18 − 2.59i)9-s + (0.0475 − 0.998i)10-s + (1.42 − 1.35i)12-s + (1.16 + 1.34i)14-s + (0.815 + 1.78i)15-s + (0.928 − 0.371i)16-s + (−0.934 + 2.69i)18-s + (0.0475 + 0.998i)20-s + (−3.24 − 1.29i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.073864390\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.073864390\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.995 - 0.0950i)T \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 67 | \( 1 + (0.327 - 0.945i)T \) |
good | 3 | \( 1 + (-1.65 + 1.06i)T + (0.415 - 0.909i)T^{2} \) |
| 7 | \( 1 + (1.03 + 1.44i)T + (-0.327 + 0.945i)T^{2} \) |
| 11 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 13 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 17 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 19 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 23 | \( 1 + (-0.888 - 0.458i)T + (0.580 + 0.814i)T^{2} \) |
| 29 | \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.154 + 0.445i)T + (-0.786 + 0.618i)T^{2} \) |
| 43 | \( 1 + (-0.428 + 0.494i)T + (-0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (-0.0688 - 1.44i)T + (-0.995 + 0.0950i)T^{2} \) |
| 53 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.223 + 0.175i)T + (0.235 - 0.971i)T^{2} \) |
| 71 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 73 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 79 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 83 | \( 1 + (-1.07 + 0.431i)T + (0.723 - 0.690i)T^{2} \) |
| 89 | \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.496763479926060930731230314038, −8.852573793700857162965965889661, −7.68820454954955578779628545997, −7.49264211710613490105836859356, −6.76796082059093656189184068485, −6.28479752346199474101115838870, −3.81587416120543448595321320708, −3.22313223592870771826412779560, −2.40561407232622758812268283252, −1.07714906912893383293377559048,
1.96584864339041134867457501641, 2.82741809109711057372450659073, 3.56712856721854168055968432628, 4.79869818095111307125041809802, 5.79753031714344636289117987391, 7.08574858737858474064847364633, 8.159691733931486176084628408747, 8.599125333167721882014994460527, 9.181968234506796259392715971823, 9.572366840995407021344081289525