| L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + 2i·13-s + 14-s + 16-s − 2·19-s − 2·26-s + i·28-s − 6·29-s + 8·31-s + i·32-s + 4i·37-s − 2i·38-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s + 0.554i·13-s + 0.267·14-s + 0.250·16-s − 0.458·19-s − 0.392·26-s + 0.188i·28-s − 1.11·29-s + 1.43·31-s + 0.176i·32-s + 0.657i·37-s − 0.324i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.333493941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.333493941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 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
−8.913588742562795045312389790636, −7.969147393638267170447910365320, −7.53524729916976273244206017927, −6.53678653285585313234974678592, −6.16171876502182176338580026076, −5.06586964490221257497733588172, −4.39717522695574308172824452433, −3.59135396476997925927508395011, −2.40296417458936995675272446746, −1.07208474722688295718326417191,
0.47436319440764993860101528223, 1.82744132002226770271992457725, 2.68763404797971428976602987156, 3.58830600477777672384790510525, 4.44042058937328080571874913564, 5.33040325595765643015636409375, 6.01343750212412150270780751421, 6.96553250866301347378844342410, 7.915321324046354594827067599451, 8.489250078825845077535406397302
