| L(s) = 1 | + 384·17-s + 116·25-s − 1.15e3·41-s − 1.34e3·49-s − 2.15e3·73-s + 5.37e3·89-s − 2.36e3·97-s − 768·113-s + 716·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.92e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 5.47·17-s + 0.927·25-s − 4.38·41-s − 3.93·49-s − 3.45·73-s + 6.40·89-s − 2.47·97-s − 0.639·113-s + 0.537·121-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 + 3.60·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.999092094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.999092094\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 674 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 358 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3962 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 12118 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 12046 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 48586 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 17810 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 12554 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 288 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 135910 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 99554 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 265306 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 180358 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 116326 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 117578 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 70610 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 538 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 30094 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 956950 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1344 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 590 T + p^{3} T^{2} )^{4} \) |
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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
−7.44045830368397926417717968862, −7.12790241791655359005298775365, −6.79159617245783805712060548315, −6.52724301217023561538961372054, −6.38454333721606107437344175009, −6.11680769966775482349963225441, −5.82504513154170655832187582176, −5.54411596445327404368004074428, −5.20804130822769830823292648588, −5.09106461932809977954996700095, −4.99569815827947358186105635984, −4.84943085230345851289757246062, −4.16132109025624225210212189058, −3.95873008401767984134499081558, −3.47459442794847392620803553764, −3.39900442035816635716869440621, −3.16845078158073063177759936667, −2.92646275616965865637226002833, −2.89376184761651871691017364408, −1.84094607005923346201328087714, −1.76066742267631387019340426061, −1.44610750842263446805586683460, −1.20012262786465060824495175795, −0.67730892587295511649770278242, −0.33754619195413391029916677792,
0.33754619195413391029916677792, 0.67730892587295511649770278242, 1.20012262786465060824495175795, 1.44610750842263446805586683460, 1.76066742267631387019340426061, 1.84094607005923346201328087714, 2.89376184761651871691017364408, 2.92646275616965865637226002833, 3.16845078158073063177759936667, 3.39900442035816635716869440621, 3.47459442794847392620803553764, 3.95873008401767984134499081558, 4.16132109025624225210212189058, 4.84943085230345851289757246062, 4.99569815827947358186105635984, 5.09106461932809977954996700095, 5.20804130822769830823292648588, 5.54411596445327404368004074428, 5.82504513154170655832187582176, 6.11680769966775482349963225441, 6.38454333721606107437344175009, 6.52724301217023561538961372054, 6.79159617245783805712060548315, 7.12790241791655359005298775365, 7.44045830368397926417717968862
