| L(s) = 1 | + 64·9-s − 24·11-s − 104·23-s + 1.72e3·25-s − 1.40e3·29-s − 3.39e3·37-s + 2.02e3·43-s − 1.66e4·53-s + 2.08e4·67-s + 1.98e3·71-s + 2.96e4·79-s + 5.79e3·81-s − 1.53e3·99-s + 4.07e3·107-s + 7.04e4·109-s + 7.87e4·113-s − 1.57e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.58e4·169-s + ⋯ |
| L(s) = 1 | + 0.790·9-s − 0.198·11-s − 0.196·23-s + 2.75·25-s − 1.67·29-s − 2.47·37-s + 1.09·43-s − 5.93·53-s + 4.63·67-s + 0.393·71-s + 4.74·79-s + 0.882·81-s − 0.156·99-s + 0.355·107-s + 5.92·109-s + 6.16·113-s − 1.07·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 1.25·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(8.438768576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.438768576\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2:C_4$ | \( 1 - 64 T^{2} - 1696 T^{4} - 64 p^{8} T^{6} + p^{16} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 1724 T^{2} + 1374142 T^{4} - 1724 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 12 T + 8100 T^{2} + 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 35804 T^{2} + 1120765054 T^{4} - 35804 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 109328 T^{2} + 11607734400 T^{4} + 109328 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 349488 T^{2} + 60105434976 T^{4} - 349488 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 52 T + 518886 T^{2} + 52 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 704 T + 1537498 T^{2} + 704 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 1723364 T^{2} + 1763753061958 T^{4} - 1723364 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 1696 T + 4464834 T^{2} + 1696 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 10204640 T^{2} + 41989966346560 T^{4} - 10204640 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 1012 T + 4332388 T^{2} - 1012 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 523468 T^{2} - 21953751403130 T^{4} + 523468 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 8340 T + 32172990 T^{2} + 8340 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 9053280 T^{2} + 308187661511040 T^{4} - 9053280 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 45572348 T^{2} + 900315190880446 T^{4} - 45572348 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 10408 T + 66148266 T^{2} - 10408 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 992 T + 47974306 T^{2} - 992 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 75977280 T^{2} + 3047966707757760 T^{4} - 75977280 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 14808 T + 122413578 T^{2} - 14808 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 95940864 T^{2} + 6613088999857824 T^{4} - 95940864 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 143773584 T^{2} + 11740110102560544 T^{4} - 143773584 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 240657696 T^{2} + 28374663059580288 T^{4} - 240657696 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−6.69088027749462480993967115484, −6.59714511164321484599541300574, −6.53820874490246058498946874773, −6.03421213471859166949727349275, −5.93554209011586964285843024726, −5.63922732860856428851850825392, −5.29437482348242038299326068045, −5.02553059523935995237408960216, −4.79958069769054424911054352004, −4.73878610927881458468485491147, −4.61866687242585342899273819597, −4.15220972280855422168972767826, −3.79504993117872991242934837098, −3.41024702136018831306803819202, −3.23877372711743745050830852176, −3.19812384611801695758997104056, −3.12395818048795462532171945189, −2.11149656247420737604142336600, −2.00848673354322839256653527170, −1.94086967683122657317748911191, −1.90182436802328393057018087007, −1.03660491008783978423766119114, −0.843133903940141995863356957210, −0.59098548470444958289279929571, −0.37414595135672097112321387059,
0.37414595135672097112321387059, 0.59098548470444958289279929571, 0.843133903940141995863356957210, 1.03660491008783978423766119114, 1.90182436802328393057018087007, 1.94086967683122657317748911191, 2.00848673354322839256653527170, 2.11149656247420737604142336600, 3.12395818048795462532171945189, 3.19812384611801695758997104056, 3.23877372711743745050830852176, 3.41024702136018831306803819202, 3.79504993117872991242934837098, 4.15220972280855422168972767826, 4.61866687242585342899273819597, 4.73878610927881458468485491147, 4.79958069769054424911054352004, 5.02553059523935995237408960216, 5.29437482348242038299326068045, 5.63922732860856428851850825392, 5.93554209011586964285843024726, 6.03421213471859166949727349275, 6.53820874490246058498946874773, 6.59714511164321484599541300574, 6.69088027749462480993967115484
