| L(s) = 1 | + 2·7-s + 3·11-s − 2·13-s + 6·17-s + 2·19-s + 6·23-s + 5·25-s + 6·29-s − 4·31-s − 8·37-s + 9·41-s − 43-s + 6·47-s + 7·49-s − 24·53-s − 3·59-s − 8·61-s + 5·67-s − 24·71-s + 22·73-s + 6·77-s − 4·79-s − 12·83-s − 12·89-s − 4·91-s − 5·97-s + 14·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.904·11-s − 0.554·13-s + 1.45·17-s + 0.458·19-s + 1.25·23-s + 25-s + 1.11·29-s − 0.718·31-s − 1.31·37-s + 1.40·41-s − 0.152·43-s + 0.875·47-s + 49-s − 3.29·53-s − 0.390·59-s − 1.02·61-s + 0.610·67-s − 2.84·71-s + 2.57·73-s + 0.683·77-s − 0.450·79-s − 1.31·83-s − 1.27·89-s − 0.419·91-s − 0.507·97-s + 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.126814041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126814041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.25579802640865048040183917587, −11.07866366555740439359771270519, −10.39754027979041133977890743745, −10.19258215256983998838518179889, −9.313408029344223610896758252675, −9.296744252914712871356596893282, −8.717328239856894343614273700497, −8.180894277460319958189576626761, −7.49243225827977902162309094611, −7.47832958005682201226848461520, −6.68599060143512744933427306861, −6.33012584887830489800912173977, −5.40398530316707544838539381833, −5.30129445065071527894209109712, −4.56105700977336424427672751606, −4.11821906729722128344203142504, −3.09992969784421783300365466870, −2.96074728033307913821773537903, −1.66912056630453280328443050680, −1.07858203020415330750621583833,
1.07858203020415330750621583833, 1.66912056630453280328443050680, 2.96074728033307913821773537903, 3.09992969784421783300365466870, 4.11821906729722128344203142504, 4.56105700977336424427672751606, 5.30129445065071527894209109712, 5.40398530316707544838539381833, 6.33012584887830489800912173977, 6.68599060143512744933427306861, 7.47832958005682201226848461520, 7.49243225827977902162309094611, 8.180894277460319958189576626761, 8.717328239856894343614273700497, 9.296744252914712871356596893282, 9.313408029344223610896758252675, 10.19258215256983998838518179889, 10.39754027979041133977890743745, 11.07866366555740439359771270519, 11.25579802640865048040183917587
