| L(s) = 1 | − 2-s + 4-s + (−1.12 − 1.93i)5-s − 8-s + (1.12 + 1.93i)10-s + 0.602i·11-s + 0.0571·13-s + 16-s − 0.347i·17-s + 4.92i·19-s + (−1.12 − 1.93i)20-s − 0.602i·22-s + 1.45·23-s + (−2.46 + 4.35i)25-s − 0.0571·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.503 − 0.863i)5-s − 0.353·8-s + (0.356 + 0.610i)10-s + 0.181i·11-s + 0.0158·13-s + 0.250·16-s − 0.0842i·17-s + 1.12i·19-s + (−0.251 − 0.431i)20-s − 0.128i·22-s + 0.304·23-s + (−0.492 + 0.870i)25-s − 0.0112·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4172931863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4172931863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.12 + 1.93i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 0.602iT - 11T^{2} \) |
| 13 | \( 1 - 0.0571T + 13T^{2} \) |
| 17 | \( 1 + 0.347iT - 17T^{2} \) |
| 19 | \( 1 - 4.92iT - 19T^{2} \) |
| 23 | \( 1 - 1.45T + 23T^{2} \) |
| 29 | \( 1 + 6.49iT - 29T^{2} \) |
| 31 | \( 1 - 5.05iT - 31T^{2} \) |
| 37 | \( 1 + 1.16iT - 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 + 7.74iT - 47T^{2} \) |
| 53 | \( 1 - 5.55T + 53T^{2} \) |
| 59 | \( 1 - 0.890T + 59T^{2} \) |
| 61 | \( 1 - 5.72iT - 61T^{2} \) |
| 67 | \( 1 + 0.110iT - 67T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 - 0.914T + 73T^{2} \) |
| 79 | \( 1 - 1.38T + 79T^{2} \) |
| 83 | \( 1 - 0.429iT - 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.190930241497415791337050777229, −7.45421211768824142264609587003, −6.80536451037931030815755243406, −5.78547466226608446911864417957, −5.17509126504289632167096449669, −4.16052654129815139604575338499, −3.48271232096207933546786350152, −2.23196647100123870760785250309, −1.29987687631689066330188325975, −0.16621391570181138101565216985,
1.16779219900289231289071395209, 2.49529281987136018206338184181, 3.07739472579100330196130892250, 4.04186399764630883995474763429, 4.98634834554654965105885592908, 6.01298126018197721654923797457, 6.72283155934415372594214324975, 7.24324152517852633018040637007, 7.976450470444512459435568579591, 8.601553759178100181077610765657
