| L(s) = 1 | − 512·4-s + 4.12e4·7-s + 8.63e5·13-s + 2.62e5·16-s + 7.51e6·19-s + 1.53e7·25-s − 2.11e7·28-s − 7.19e7·31-s + 5.78e7·37-s + 3.45e8·43-s + 7.12e8·49-s − 4.41e8·52-s − 2.60e9·61-s − 1.34e8·64-s − 3.63e9·67-s + 3.88e9·73-s − 3.84e9·76-s − 4.57e9·79-s + 3.56e10·91-s + 6.79e9·97-s − 7.86e9·100-s − 2.32e10·103-s − 2.35e10·109-s + 1.08e10·112-s + 5.06e10·121-s + 3.68e10·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2.45·7-s + 2.32·13-s + 1/4·16-s + 3.03·19-s + 1.57·25-s − 1.22·28-s − 2.51·31-s + 0.834·37-s + 2.35·43-s + 2.52·49-s − 1.16·52-s − 3.08·61-s − 1/8·64-s − 2.69·67-s + 1.87·73-s − 1.51·76-s − 1.48·79-s + 5.70·91-s + 0.791·97-s − 0.786·100-s − 2.00·103-s − 1.52·109-s + 0.613·112-s + 1.95·121-s + 1.25·124-s + 7.44·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.897831905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.897831905\) |
| \(L(6)\) |
|
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 | $C_2$ | \( 1 + p^{9} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 15366752 T^{2} + p^{20} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2948 p T + p^{10} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 50675133074 T^{2} + p^{20} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 431528 T + p^{10} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1791552688864 T^{2} + p^{20} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3755504 T + p^{10} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20900948558210 T^{2} + p^{20} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 254820400418624 T^{2} + p^{20} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 35971636 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 28933886 T + p^{10} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 16339184026436384 T^{2} + p^{20} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 172966040 T + p^{10} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 77160642453412898 T^{2} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 45068830353920 p^{2} T^{2} + p^{20} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1001689819439661170 T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 1301992750 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 1816668472 T + p^{10} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6464819094775259330 T^{2} + p^{20} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1944213104 T + p^{10} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2287819756 T + p^{10} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 5393424130005052174 T^{2} + p^{20} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 24835242107390980160 T^{2} + p^{20} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 3397569776 T + p^{10} T^{2} )^{2} \) |
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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
−16.42168057829120486866279571358, −16.13023381239634469341055157669, −15.13226978059383799939929522385, −14.50171150607836457620590223909, −13.95412654283183563499739429909, −13.58090058998378163377144126537, −12.53921233195702708810577056144, −11.63683158058017083412006182635, −10.94895906106942296794150535991, −10.86310851021769236801725480511, −9.197446664254483966511723861452, −8.886225961600291623605582995975, −7.82580070873139726363472237347, −7.50467218054682081435940226998, −5.81053167691763795934386788277, −5.20747942694558523181225509787, −4.30417830550373637449004775770, −3.21904344311730871200616010403, −1.28376183735305365616209894928, −1.26365775907379419790499223471,
1.26365775907379419790499223471, 1.28376183735305365616209894928, 3.21904344311730871200616010403, 4.30417830550373637449004775770, 5.20747942694558523181225509787, 5.81053167691763795934386788277, 7.50467218054682081435940226998, 7.82580070873139726363472237347, 8.886225961600291623605582995975, 9.197446664254483966511723861452, 10.86310851021769236801725480511, 10.94895906106942296794150535991, 11.63683158058017083412006182635, 12.53921233195702708810577056144, 13.58090058998378163377144126537, 13.95412654283183563499739429909, 14.50171150607836457620590223909, 15.13226978059383799939929522385, 16.13023381239634469341055157669, 16.42168057829120486866279571358
