| L(s) = 1 | + 1.53·5-s + (−0.414 + 2.61i)7-s + 0.585i·11-s + 2.16i·13-s + 5.86·17-s + 5.22i·19-s − 2.24i·23-s − 2.65·25-s − 5.41i·29-s + 4.32i·31-s + (−0.634 + 4i)35-s + 4·37-s + 8.92·41-s + 10.4·43-s − 7.39·47-s + ⋯ |
| L(s) = 1 | + 0.684·5-s + (−0.156 + 0.987i)7-s + 0.176i·11-s + 0.600i·13-s + 1.42·17-s + 1.19i·19-s − 0.467i·23-s − 0.531·25-s − 1.00i·29-s + 0.777i·31-s + (−0.107 + 0.676i)35-s + 0.657·37-s + 1.39·41-s + 1.59·43-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44591 + 0.609714i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44591 + 0.609714i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.414 - 2.61i)T \) |
| good | 5 | \( 1 - 1.53T + 5T^{2} \) |
| 11 | \( 1 - 0.585iT - 11T^{2} \) |
| 13 | \( 1 - 2.16iT - 13T^{2} \) |
| 17 | \( 1 - 5.86T + 17T^{2} \) |
| 19 | \( 1 - 5.22iT - 19T^{2} \) |
| 23 | \( 1 + 2.24iT - 23T^{2} \) |
| 29 | \( 1 + 5.41iT - 29T^{2} \) |
| 31 | \( 1 - 4.32iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 - 5.41iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 + 4.58iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 5.86T + 89T^{2} \) |
| 97 | \( 1 + 8.28iT - 97T^{2} \) |
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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
−11.01160076366881497201641082072, −9.883328812720841227692090206803, −9.480490193053696064187552268151, −8.388700969082277697411505643034, −7.48519467089852407714226736857, −6.05673309105898376037676304251, −5.74600863134440331842046145546, −4.34820089842539837856802372495, −2.93197064715842146159618614586, −1.72229949147061915538459039038,
1.05002571014374734811656084089, 2.78548123072421221440354498562, 3.96195373907861207408160276652, 5.24861189458954290962907687383, 6.10836275012038247210031856439, 7.25686698592866904824710752731, 7.942459634311527474472667412529, 9.267858308491782322869870806010, 9.904465635815059330369619122814, 10.72265364251145717634299447765
