| L(s) = 1 | − 115·16-s + 1.80e3·43-s + 1.37e3·49-s + 3.39e3·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 1.79·16-s + 6.41·43-s + 4·49-s + 3.24·103-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + 0.000270·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.366005925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.366005925\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 115 T^{4} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 3742 T^{4} + p^{12} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 11 | $D_4\times C_2$ | \( 1 - 1486370 T^{4} + p^{12} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 315317810 T^{4} + p^{12} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 452 T + p^{3} T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 19480835090 T^{4} + p^{12} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 78746477470 T^{4} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 438410 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 234490873970 T^{4} + p^{12} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 811150 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 633473875010 T^{4} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 926581329650 T^{4} + p^{12} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
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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
−9.408438562124640929267531943094, −9.110946984540183029243930824246, −8.973774865918991405321458958340, −8.735400857083690430728815360239, −8.561013985526855336965885325755, −7.74799160357728494602927486165, −7.70088033977789600764664986507, −7.52620405642343953344572689283, −7.05361483036729596833337850764, −7.02740685829004620806370228726, −6.43930566248715737132062517212, −6.02761560678818687588752145036, −5.97098725603595227383763173865, −5.46765207885993771344812385107, −5.30028361417664673156297147422, −4.63880727924482051354889512484, −4.28916520528875930588625783628, −4.08861906623796825448220164795, −3.93244288367769326776970706608, −3.07536055520435076079127533659, −2.57712972834261045861582992102, −2.39050690219638523959115644465, −1.93414730410789057295757879670, −0.74947780079364504799971857884, −0.72804117484882406288662974146,
0.72804117484882406288662974146, 0.74947780079364504799971857884, 1.93414730410789057295757879670, 2.39050690219638523959115644465, 2.57712972834261045861582992102, 3.07536055520435076079127533659, 3.93244288367769326776970706608, 4.08861906623796825448220164795, 4.28916520528875930588625783628, 4.63880727924482051354889512484, 5.30028361417664673156297147422, 5.46765207885993771344812385107, 5.97098725603595227383763173865, 6.02761560678818687588752145036, 6.43930566248715737132062517212, 7.02740685829004620806370228726, 7.05361483036729596833337850764, 7.52620405642343953344572689283, 7.70088033977789600764664986507, 7.74799160357728494602927486165, 8.561013985526855336965885325755, 8.735400857083690430728815360239, 8.973774865918991405321458958340, 9.110946984540183029243930824246, 9.408438562124640929267531943094
