| L(s) = 1 | + 1.93i·5-s − 0.732i·7-s − 5.27·11-s − 4.46·13-s + 0.896i·17-s + 1.26i·19-s + 1.41·23-s + 1.26·25-s + 5.41i·29-s + 7.46i·31-s + 1.41·35-s + 7.73·37-s + 0.378i·41-s − 8.73i·43-s − 4.62·47-s + ⋯ |
| L(s) = 1 | + 0.863i·5-s − 0.276i·7-s − 1.59·11-s − 1.23·13-s + 0.217i·17-s + 0.290i·19-s + 0.294·23-s + 0.253·25-s + 1.00i·29-s + 1.34i·31-s + 0.239·35-s + 1.27·37-s + 0.0591i·41-s − 1.33i·43-s − 0.674·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5710973038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5710973038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.93iT - 5T^{2} \) |
| 7 | \( 1 + 0.732iT - 7T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 - 0.896iT - 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 5.41iT - 29T^{2} \) |
| 31 | \( 1 - 7.46iT - 31T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 - 0.378iT - 41T^{2} \) |
| 43 | \( 1 + 8.73iT - 43T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 - 2.44iT - 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 16.5iT - 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 14.9iT - 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−7.85155042778538378214552674900, −7.29311192044915748407702318668, −6.81810015556101554675259173954, −5.81539168441031806944151824212, −5.09714194713949994223048352881, −4.46008518211380205125970409875, −3.16398766667903062052870269044, −2.83370462277742226236326965543, −1.78661209937616048470810355157, −0.17443457264764364745125563895,
0.914287708961009985560697375064, 2.43542479076989545663079567764, 2.69620839392640509349813127375, 4.15718737985963089042055528052, 4.83683963724771038217674325881, 5.33841456129718193089704854224, 6.08098176143403896796170238909, 7.11312823872764077924665940238, 7.86461758074977829476975954578, 8.172894433610833888986703155935
