| L(s) = 1 | − 5-s − 8·7-s + 6·11-s − 6·13-s + 2·17-s + 7·19-s + 4·23-s + 29-s − 10·31-s + 8·35-s − 8·37-s + 2·41-s + 6·47-s + 34·49-s − 6·53-s − 6·55-s − 59-s + 7·61-s + 6·65-s + 14·67-s + 15·71-s − 12·73-s − 48·77-s + 79-s − 32·83-s − 2·85-s + 17·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 3.02·7-s + 1.80·11-s − 1.66·13-s + 0.485·17-s + 1.60·19-s + 0.834·23-s + 0.185·29-s − 1.79·31-s + 1.35·35-s − 1.31·37-s + 0.312·41-s + 0.875·47-s + 34/7·49-s − 0.824·53-s − 0.809·55-s − 0.130·59-s + 0.896·61-s + 0.744·65-s + 1.71·67-s + 1.78·71-s − 1.40·73-s − 5.47·77-s + 0.112·79-s − 3.51·83-s − 0.216·85-s + 1.80·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7551122082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7551122082\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 17 T + 200 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 47 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.966891459658000588404092848485, −8.651901881445845856311804418725, −7.88049762224306117566402467395, −7.45990197158958493512580376178, −7.24937173782019793001168084143, −6.83641007431330357023185711493, −6.68645229352955628743311182619, −6.38226162138481808338497080014, −5.76357538241750172154811519651, −5.36841780246809574208281192990, −5.20924209378996784352518722786, −4.43717947030611179409087808746, −3.81358153883132156076803315015, −3.73002428415643947672550827682, −3.33588375185845154265011554385, −2.88007166650567570647748443265, −2.58118126271911375319498048089, −1.70956062175205331654767734095, −0.984050751173737153450723924121, −0.31317974222939298303837249091,
0.31317974222939298303837249091, 0.984050751173737153450723924121, 1.70956062175205331654767734095, 2.58118126271911375319498048089, 2.88007166650567570647748443265, 3.33588375185845154265011554385, 3.73002428415643947672550827682, 3.81358153883132156076803315015, 4.43717947030611179409087808746, 5.20924209378996784352518722786, 5.36841780246809574208281192990, 5.76357538241750172154811519651, 6.38226162138481808338497080014, 6.68645229352955628743311182619, 6.83641007431330357023185711493, 7.24937173782019793001168084143, 7.45990197158958493512580376178, 7.88049762224306117566402467395, 8.651901881445845856311804418725, 8.966891459658000588404092848485
