| L(s) = 1 | + 2·9-s − 136·19-s − 56·31-s + 188·49-s − 184·61-s − 88·79-s − 77·81-s − 344·109-s + 124·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 548·169-s − 272·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2/9·9-s − 7.15·19-s − 1.80·31-s + 3.83·49-s − 3.01·61-s − 1.11·79-s − 0.950·81-s − 3.15·109-s + 1.02·121-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 + 3.24·169-s − 1.59·171-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0001308061579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0001308061579\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 398 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 562 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 398 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2642 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3634 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2798 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3998 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6782 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 7954 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7202 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 578 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 3934 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.66638928993202976700456397516, −6.61146285672673865505168395977, −6.44596269020314768667014101283, −6.05349648417671664937376774984, −5.89911202009456025601811472563, −5.79720231659006233478279235949, −5.46641954491667331468390487436, −5.33536529099693282095377787580, −4.69707560066858465016458093595, −4.63860788961005539345229045868, −4.48244269934000413310430976965, −4.16656189511005046018473942612, −3.92890121185152428876901230082, −3.91128773101413937118617238929, −3.86263442570371877847876657115, −3.01370392120000591473021930520, −2.89535148860365289166359237299, −2.63878534003948261062056868727, −2.22765364733967551678613987539, −2.00793518363584693449368993657, −1.86220032734003957367456191315, −1.67402543037542901523687387110, −1.04131617643235049155565538138, −0.42670323305732760533765521618, −0.00247494265833574726372291768,
0.00247494265833574726372291768, 0.42670323305732760533765521618, 1.04131617643235049155565538138, 1.67402543037542901523687387110, 1.86220032734003957367456191315, 2.00793518363584693449368993657, 2.22765364733967551678613987539, 2.63878534003948261062056868727, 2.89535148860365289166359237299, 3.01370392120000591473021930520, 3.86263442570371877847876657115, 3.91128773101413937118617238929, 3.92890121185152428876901230082, 4.16656189511005046018473942612, 4.48244269934000413310430976965, 4.63860788961005539345229045868, 4.69707560066858465016458093595, 5.33536529099693282095377787580, 5.46641954491667331468390487436, 5.79720231659006233478279235949, 5.89911202009456025601811472563, 6.05349648417671664937376774984, 6.44596269020314768667014101283, 6.61146285672673865505168395977, 6.66638928993202976700456397516
