| L(s) = 1 | + 3-s + 3·5-s + 9-s − 4·11-s − 3·13-s + 3·15-s + 4·17-s + 23-s + 4·25-s + 27-s − 3·29-s + 6·31-s − 4·33-s + 9·37-s − 3·39-s − 9·41-s + 3·43-s + 3·45-s + 7·47-s + 4·51-s + 4·53-s − 12·55-s + 6·59-s + 10·61-s − 9·65-s − 4·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1/3·9-s − 1.20·11-s − 0.832·13-s + 0.774·15-s + 0.970·17-s + 0.208·23-s + 4/5·25-s + 0.192·27-s − 0.557·29-s + 1.07·31-s − 0.696·33-s + 1.47·37-s − 0.480·39-s − 1.40·41-s + 0.457·43-s + 0.447·45-s + 1.02·47-s + 0.560·51-s + 0.549·53-s − 1.61·55-s + 0.781·59-s + 1.28·61-s − 1.11·65-s − 0.488·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.551563452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.551563452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.14005943086366, −12.67527572157008, −12.09492963762663, −11.74988356284997, −10.90124945064626, −10.54184551361067, −9.948498796639974, −9.854057117888675, −9.382097021585106, −8.756459509724438, −8.284515192222813, −7.679398029789917, −7.467335100498024, −6.737647960366815, −6.241776354948295, −5.653958003338686, −5.230303452627287, −4.903844168455591, −4.130740829307484, −3.492282181228748, −2.756843517271680, −2.478816870470403, −2.034315408053151, −1.229484676858967, −0.5756726106669917,
0.5756726106669917, 1.229484676858967, 2.034315408053151, 2.478816870470403, 2.756843517271680, 3.492282181228748, 4.130740829307484, 4.903844168455591, 5.230303452627287, 5.653958003338686, 6.241776354948295, 6.737647960366815, 7.467335100498024, 7.679398029789917, 8.284515192222813, 8.756459509724438, 9.382097021585106, 9.854057117888675, 9.948498796639974, 10.54184551361067, 10.90124945064626, 11.74988356284997, 12.09492963762663, 12.67527572157008, 13.14005943086366
