| L(s) = 1 | + 14·9-s − 8·19-s − 88·31-s + 124·49-s − 344·61-s − 40·79-s + 115·81-s − 40·109-s + 420·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 476·169-s − 112·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 14/9·9-s − 0.421·19-s − 2.83·31-s + 2.53·49-s − 5.63·61-s − 0.506·79-s + 1.41·81-s − 0.366·109-s + 3.47·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.81·169-s − 0.654·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.133841336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.133841336\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 14 T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 930 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1394 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2702 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2210 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 3026 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1746 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 1554 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 86 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8974 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 5406 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 3934 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 8370 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 14690 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 9982 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42724022749470430679703692835, −7.23571903658230753423736447508, −7.05683589524443391256804228604, −6.99634577536451139411412999470, −6.48856438600064249048166626747, −6.26703909058372750621813158957, −6.03531516031922481125087975138, −5.70094313383034271346327683029, −5.55995380936719639822313868625, −5.45253477416675088630634830017, −4.89892493026588767875287178916, −4.64059766464650956975778291724, −4.35981133518201608654748527092, −4.30540344276289016548231640986, −4.11667184887025202256100365308, −3.46652387914952002997261406309, −3.43977111333285076561745692151, −3.14795518883202794145402277432, −2.75470745528014728429075204654, −2.24331870576836351733297986305, −1.88615368310800691537328835268, −1.68252479635135121245549393786, −1.44667960274809516625508550828, −0.73176059322238217223091331104, −0.35808482985459704872500789857,
0.35808482985459704872500789857, 0.73176059322238217223091331104, 1.44667960274809516625508550828, 1.68252479635135121245549393786, 1.88615368310800691537328835268, 2.24331870576836351733297986305, 2.75470745528014728429075204654, 3.14795518883202794145402277432, 3.43977111333285076561745692151, 3.46652387914952002997261406309, 4.11667184887025202256100365308, 4.30540344276289016548231640986, 4.35981133518201608654748527092, 4.64059766464650956975778291724, 4.89892493026588767875287178916, 5.45253477416675088630634830017, 5.55995380936719639822313868625, 5.70094313383034271346327683029, 6.03531516031922481125087975138, 6.26703909058372750621813158957, 6.48856438600064249048166626747, 6.99634577536451139411412999470, 7.05683589524443391256804228604, 7.23571903658230753423736447508, 7.42724022749470430679703692835
