| L(s) = 1 | − 24·2-s + 304·4-s − 2.30e3·8-s + 1.31e3·9-s + 2.95e3·11-s + 5.37e3·16-s − 3.14e4·18-s − 7.09e4·22-s − 1.18e4·23-s − 1.82e3·25-s + 7.95e3·29-s + 1.29e5·32-s + 3.98e5·36-s − 1.23e5·37-s − 3.48e4·43-s + 8.99e5·44-s + 2.83e5·46-s + 4.38e4·50-s − 1.21e5·53-s − 1.90e5·58-s − 2.26e6·64-s − 5.37e5·67-s + 2.03e5·71-s − 3.02e6·72-s + 2.95e6·74-s + 7.24e5·79-s + 1.18e6·81-s + ⋯ |
| L(s) = 1 | − 3·2-s + 19/4·4-s − 9/2·8-s + 1.79·9-s + 2.22·11-s + 1.31·16-s − 5.39·18-s − 6.66·22-s − 0.971·23-s − 0.116·25-s + 0.326·29-s + 3.93·32-s + 8.54·36-s − 2.43·37-s − 0.438·43-s + 10.5·44-s + 2.91·46-s + 0.350·50-s − 0.812·53-s − 0.978·58-s − 8.64·64-s − 1.78·67-s + 0.569·71-s − 8.09·72-s + 7.29·74-s + 1.46·79-s + 2.23·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.5830073812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5830073812\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + 3 p^{2} T + p^{6} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 437 p T^{2} + p^{12} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 73 p^{2} T^{2} + p^{12} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 1479 T + p^{6} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 9418418 T^{2} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 39160991 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 46832879 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 5913 T + p^{6} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3978 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1610771759 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 61577 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2726428318 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 17414 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 20618242031 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 60513 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 37822212479 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 76554740159 T^{2} + p^{12} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 268777 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 101922 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 201769475231 T^{2} + p^{12} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 362231 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 606885669938 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 787099021441 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 626523106942 T^{2} + p^{12} T^{4} \) |
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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
−15.04010621949871852735246434324, −14.00513791113223804225368107722, −13.67938125732688828552228251766, −12.57223016689697799374055282623, −11.96354124776651058735061953977, −11.45994244414498558376313660711, −10.62326965612425487298286265025, −10.15672681276811133704987700446, −9.809314670971383227725716319706, −9.047036786990555317737329427627, −8.940711179047486676003580615923, −8.003749362503105504736183812476, −7.44319370427491552486131129137, −6.74761412627232764767027227554, −6.50466424085863227894315048370, −4.62122566597817528276383066701, −3.80103635563573510036716314498, −1.71501057957441228679450528033, −1.59674486255865772407358992621, −0.61285274376595131798757964622,
0.61285274376595131798757964622, 1.59674486255865772407358992621, 1.71501057957441228679450528033, 3.80103635563573510036716314498, 4.62122566597817528276383066701, 6.50466424085863227894315048370, 6.74761412627232764767027227554, 7.44319370427491552486131129137, 8.003749362503105504736183812476, 8.940711179047486676003580615923, 9.047036786990555317737329427627, 9.809314670971383227725716319706, 10.15672681276811133704987700446, 10.62326965612425487298286265025, 11.45994244414498558376313660711, 11.96354124776651058735061953977, 12.57223016689697799374055282623, 13.67938125732688828552228251766, 14.00513791113223804225368107722, 15.04010621949871852735246434324
