| L(s) = 1 | − 27·9-s − 140·13-s + 250·25-s − 220·37-s − 286·49-s − 364·61-s − 2.38e3·73-s + 729·81-s + 2.66e3·97-s − 1.29e3·109-s + 3.78e3·117-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s − 2.98·13-s + 2·25-s − 0.977·37-s − 0.833·49-s − 0.764·61-s − 3.81·73-s + 81-s + 2.78·97-s − 1.13·109-s + 2.98·117-s − 2·121-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 + 4.69·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8440388813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8440388813\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )( 1 + 20 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )( 1 + 56 T + p^{3} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 110 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 520 T + p^{3} T^{2} )( 1 + 520 T + p^{3} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 880 T + p^{3} T^{2} )( 1 + 880 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1190 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )( 1 + 884 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )^{2} \) |
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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
−12.11404556454237517898715750579, −11.97080426651678527172442316758, −11.52925075223636367977274557246, −10.72874747690761892007084898356, −10.35749313763207394316265647467, −9.959624652498773104774836292666, −9.103640294605888181883488790002, −9.090726508523707183617641411658, −8.288385667591260325808733910968, −7.66425583564609232077092167683, −7.21140623894425760295039773128, −6.75966867381010102619316302029, −6.01083526005726888950259177745, −5.15931570002782166839203250713, −4.99119024932835093425875404787, −4.29923251272244425073223636634, −2.94909743387969823483662727294, −2.88471476683056850439371472394, −1.84805953450394109073391430639, −0.38550567449165372010776371804,
0.38550567449165372010776371804, 1.84805953450394109073391430639, 2.88471476683056850439371472394, 2.94909743387969823483662727294, 4.29923251272244425073223636634, 4.99119024932835093425875404787, 5.15931570002782166839203250713, 6.01083526005726888950259177745, 6.75966867381010102619316302029, 7.21140623894425760295039773128, 7.66425583564609232077092167683, 8.288385667591260325808733910968, 9.090726508523707183617641411658, 9.103640294605888181883488790002, 9.959624652498773104774836292666, 10.35749313763207394316265647467, 10.72874747690761892007084898356, 11.52925075223636367977274557246, 11.97080426651678527172442316758, 12.11404556454237517898715750579
