| L(s) = 1 | − 3-s + 4·5-s + 3·9-s + 6·11-s − 2·13-s − 4·15-s − 2·17-s + 19-s − 6·23-s + 5·25-s − 8·27-s + 4·29-s − 20·31-s − 6·33-s + 4·37-s + 2·39-s − 9·41-s + 4·43-s + 12·45-s + 12·47-s − 14·49-s + 2·51-s + 2·53-s + 24·55-s − 57-s + 59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 9-s + 1.80·11-s − 0.554·13-s − 1.03·15-s − 0.485·17-s + 0.229·19-s − 1.25·23-s + 25-s − 1.53·27-s + 0.742·29-s − 3.59·31-s − 1.04·33-s + 0.657·37-s + 0.320·39-s − 1.40·41-s + 0.609·43-s + 1.78·45-s + 1.75·47-s − 2·49-s + 0.280·51-s + 0.274·53-s + 3.23·55-s − 0.132·57-s + 0.130·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.411782745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.411782745\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−13.17559442800242844336781632930, −12.80872177488541370759005044967, −12.24260835359046956467102361945, −11.82017066097331433762775052614, −11.17747823475709044203057478819, −10.76366086241662288591788528590, −9.913350056785686016007921169558, −9.833468482533578817535924151699, −9.222149647637466300986592325380, −9.014542192322780025316968967896, −7.950471377000123649883514450427, −7.19322731708696441187712211202, −6.77781396778814933241655013790, −6.15724638235657303043254216327, −5.71448302112860720533367886653, −5.16768123061812199845133429546, −4.19289057009413027612349126150, −3.67601781337916088431890643150, −2.05913040924601321064290812247, −1.67291575636457133446315204327,
1.67291575636457133446315204327, 2.05913040924601321064290812247, 3.67601781337916088431890643150, 4.19289057009413027612349126150, 5.16768123061812199845133429546, 5.71448302112860720533367886653, 6.15724638235657303043254216327, 6.77781396778814933241655013790, 7.19322731708696441187712211202, 7.950471377000123649883514450427, 9.014542192322780025316968967896, 9.222149647637466300986592325380, 9.833468482533578817535924151699, 9.913350056785686016007921169558, 10.76366086241662288591788528590, 11.17747823475709044203057478819, 11.82017066097331433762775052614, 12.24260835359046956467102361945, 12.80872177488541370759005044967, 13.17559442800242844336781632930
