| L(s) = 1 | + (1.44 − 2.62i)3-s + (−2.67 + 4.22i)5-s + 1.30i·7-s + (−4.83 − 7.59i)9-s − 9.69·11-s + 15.3·13-s + (7.23 + 13.1i)15-s + 9.62·17-s + 9.66i·19-s + (3.41 + 1.87i)21-s − 16.1·23-s + (−10.6 − 22.6i)25-s + (−26.9 + 1.74i)27-s + 47.2·29-s + 40.6·31-s + ⋯ |
| L(s) = 1 | + (0.481 − 0.876i)3-s + (−0.535 + 0.844i)5-s + 0.185i·7-s + (−0.536 − 0.843i)9-s − 0.881·11-s + 1.18·13-s + (0.482 + 0.875i)15-s + 0.566·17-s + 0.508i·19-s + (0.162 + 0.0893i)21-s − 0.703·23-s + (−0.426 − 0.904i)25-s + (−0.997 + 0.0644i)27-s + 1.63·29-s + 1.31·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.978716759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.978716759\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.44 + 2.62i)T \) |
| 5 | \( 1 + (2.67 - 4.22i)T \) |
| good | 7 | \( 1 - 1.30iT - 49T^{2} \) |
| 11 | \( 1 + 9.69T + 121T^{2} \) |
| 13 | \( 1 - 15.3T + 169T^{2} \) |
| 17 | \( 1 - 9.62T + 289T^{2} \) |
| 19 | \( 1 - 9.66iT - 361T^{2} \) |
| 23 | \( 1 + 16.1T + 529T^{2} \) |
| 29 | \( 1 - 47.2T + 841T^{2} \) |
| 31 | \( 1 - 40.6T + 961T^{2} \) |
| 37 | \( 1 - 43.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 4.20iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 53.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 48.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 9.87iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 70.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 92.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 25.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 41.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 17.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 92.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 54.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 102. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 97.2iT - 9.40e3T^{2} \) |
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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
−9.909371265511768760638748707846, −8.585094444467909795230836261356, −8.093124780649024024180948302248, −7.40942422540168583443696360022, −6.38854464841579132344630020167, −5.83753402429086344917541474146, −4.22907665962622317326825868196, −3.17712252568937621039890259159, −2.42653959991043057138484181928, −0.901608264381811161353886858989,
0.849694031806492382545851938905, 2.59191363496711126450638688060, 3.67675627469646640722355764765, 4.49854129473033947835665964297, 5.25764580750179863536009308120, 6.31455165739453583279270084224, 7.84585328094107123217482620303, 8.174676069919792627434739387443, 8.975948094851626095098000579944, 9.860114278216559043490321907445
