| L(s) = 1 | − 243·3-s + 512·4-s − 1.56e4·7-s + 3.93e4·9-s − 1.24e5·12-s − 3.15e4·13-s + 9.76e5·19-s + 3.80e6·21-s − 1.95e6·25-s − 4.78e6·27-s − 8.01e6·28-s + 2.01e7·36-s + 7.67e6·39-s + 2.77e7·43-s + 1.03e8·49-s − 1.61e7·52-s − 2.37e8·57-s − 9.80e7·61-s − 6.16e8·63-s − 1.34e8·64-s + 4.37e8·67-s − 1.84e8·73-s + 4.74e8·75-s + 5.00e8·76-s + 1.19e9·79-s + 3.87e8·81-s + 1.94e9·84-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 4-s − 2.46·7-s + 2·9-s − 1.73·12-s − 0.306·13-s + 1.71·19-s + 4.26·21-s − 25-s − 1.73·27-s − 2.46·28-s + 2·36-s + 0.531·39-s + 1.23·43-s + 2.55·49-s − 0.306·52-s − 2.97·57-s − 0.906·61-s − 4.92·63-s − 64-s + 2.65·67-s − 0.760·73-s + 1.73·75-s + 1.71·76-s + 3.45·79-s + 81-s + 4.26·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6001744722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6001744722\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{5} T + p^{9} T^{2} \) |
| 19 | $C_2$ | \( 1 - 976696 T + p^{9} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p^{9} T^{2} + p^{18} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 7829 T + p^{9} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 86777 T + p^{9} T^{2} )( 1 + 118370 T + p^{9} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p^{9} T^{2} + p^{18} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p^{9} T^{2} + p^{18} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7939241 T + p^{9} T^{2} )( 1 + 7939241 T + p^{9} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6880789 T + p^{9} T^{2} )( 1 + 6880789 T + p^{9} T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 44367317 T + p^{9} T^{2} )( 1 + 16577080 T + p^{9} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p^{9} T^{2} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 117903058 T + p^{9} T^{2} )( 1 + 215975699 T + p^{9} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 324823339 T + p^{9} T^{2} )( 1 - 112542320 T + p^{9} T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 296368310 T + p^{9} T^{2} )( 1 + 480959047 T + p^{9} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 616732324 T + p^{9} T^{2} )( 1 - 580920227 T + p^{9} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 372733541 T + p^{9} T^{2} )( 1 + 1661661811 T + p^{9} T^{2} ) \) |
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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
−13.49964686470248674071385462455, −12.69343615539276038784406373624, −12.19096301021078688615588365108, −12.09931225045547378960535901493, −11.16151538963473168653615109243, −10.94315409830250265999285602584, −10.07397269224408230376241003339, −9.615111296069645716728044563040, −9.392803008146507813752950697527, −7.85879796324620778338816844789, −7.02740813043268387651911947812, −6.85700788209995593929391627478, −6.11334904115822749487478019546, −5.81619074649874957453237445884, −4.98882645662554370317699323420, −3.85106554334753250899310433681, −3.16396657446863793702191739907, −2.29324114891352498085098225679, −1.06634177622684301320919154011, −0.31426759905264927185373039777,
0.31426759905264927185373039777, 1.06634177622684301320919154011, 2.29324114891352498085098225679, 3.16396657446863793702191739907, 3.85106554334753250899310433681, 4.98882645662554370317699323420, 5.81619074649874957453237445884, 6.11334904115822749487478019546, 6.85700788209995593929391627478, 7.02740813043268387651911947812, 7.85879796324620778338816844789, 9.392803008146507813752950697527, 9.615111296069645716728044563040, 10.07397269224408230376241003339, 10.94315409830250265999285602584, 11.16151538963473168653615109243, 12.09931225045547378960535901493, 12.19096301021078688615588365108, 12.69343615539276038784406373624, 13.49964686470248674071385462455
