L(s) = 1 | + 2.23·3-s + 2.82i·7-s + 2.00·9-s − 2.23i·11-s − 6.32·13-s + 5i·17-s − 2.23i·19-s + 6.32i·21-s + 5.65i·23-s − 2.23·27-s + 6.32i·29-s − 5.00i·33-s + 6.32·37-s − 14.1·39-s + 3·41-s + ⋯ |
L(s) = 1 | + 1.29·3-s + 1.06i·7-s + 0.666·9-s − 0.674i·11-s − 1.75·13-s + 1.21i·17-s − 0.512i·19-s + 1.38i·21-s + 1.17i·23-s − 0.430·27-s + 1.17i·29-s − 0.870i·33-s + 1.03·37-s − 2.26·39-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.818197482\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.818197482\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 6.32T + 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 + 2.23iT - 19T^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 - 6.32iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 6.32iT - 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 15iT - 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.848844039053204407340831723220, −8.260288592742287225397149893481, −7.65296793846013678272128499859, −6.81818572591556925250557465657, −5.77574845921972377505074004413, −5.14501154777158603977162081930, −4.06234848963603895044356546691, −3.07667112476848635233065951633, −2.58471902532718894014288085888, −1.64753644880678847841600648491,
0.41998611454189192793924161490, 2.04834286343848006680247694264, 2.65050762315209851037048858378, 3.59695025774478556589644183123, 4.50611977574596961318556201190, 5.01321173118266116059283128045, 6.47041763009799028563536188219, 7.15552156836560163866288746096, 7.81439420334123427102916799251, 8.156651600262471853383748042077