| L(s) = 1 | − 2.73i·5-s + 7-s + 5.46i·11-s − 6.73i·13-s − 2·17-s + 1.26i·19-s + 3.46·23-s − 2.46·25-s − 1.46i·29-s + 4·31-s − 2.73i·35-s − 1.46i·37-s + 2·41-s − 5.46i·43-s − 2.92·47-s + ⋯ |
| L(s) = 1 | − 1.22i·5-s + 0.377·7-s + 1.64i·11-s − 1.86i·13-s − 0.485·17-s + 0.290i·19-s + 0.722·23-s − 0.492·25-s − 0.271i·29-s + 0.718·31-s − 0.461i·35-s − 0.240i·37-s + 0.312·41-s − 0.833i·43-s − 0.427·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669777482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669777482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 2.73iT - 5T^{2} \) |
| 11 | \( 1 - 5.46iT - 11T^{2} \) |
| 13 | \( 1 + 6.73iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 1.46iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 1.46iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 5.46iT - 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 9.66iT - 59T^{2} \) |
| 61 | \( 1 - 11.1iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 5.66iT - 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−8.245192828735829665047939038327, −7.63673155158021827990952866846, −6.88793022317522245818562911560, −5.82156660950061288971780459035, −5.00456195242951721897687773167, −4.74056694499241160284600486528, −3.71670805153783638711646739688, −2.54909342886205905060622449962, −1.57477341267438928231811508174, −0.51308146535668103823880411817,
1.21530835295068399392295546509, 2.45366247578074513418746753260, 3.11040902021851877959284452861, 4.03615721171754609322479084351, 4.82523378632418391122968503421, 5.93321327675320842843106661936, 6.52938188789630273184078492768, 7.01874081927733690414489485105, 7.890053288870761640173581532410, 8.782161810648413425314382685575
