| L(s) = 1 | − 4·4-s + 78·13-s + 16·16-s + 92·17-s + 224·23-s + 186·25-s + 340·29-s − 184·43-s + 490·49-s − 312·52-s − 1.11e3·53-s + 1.80e3·61-s − 64·64-s − 368·68-s + 720·79-s − 896·92-s − 744·100-s − 1.24e3·101-s − 1.90e3·103-s + 472·107-s + 2.52e3·113-s − 1.36e3·116-s + 1.76e3·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.66·13-s + 1/4·16-s + 1.31·17-s + 2.03·23-s + 1.48·25-s + 2.17·29-s − 0.652·43-s + 10/7·49-s − 0.832·52-s − 2.89·53-s + 3.78·61-s − 1/8·64-s − 0.656·68-s + 1.02·79-s − 1.01·92-s − 0.743·100-s − 1.22·101-s − 1.82·103-s + 0.426·107-s + 2.10·113-s − 1.08·116-s + 1.32·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.061781957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.061781957\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 6 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 186 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1762 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 9362 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 112 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 170 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 47482 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 101290 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6558 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 194650 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 558 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 405282 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 902 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 184210 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 149078 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 85810 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1111890 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1368322 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 179710 T^{2} + p^{6} T^{4} \) |
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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
−12.05037237374991078673053131079, −11.37096372431685723576425127635, −10.96070596289974715157243391367, −10.63816751645279271955505185667, −9.975510360396630275980711510891, −9.638518538297378943700459105380, −8.788271752617548098974140432651, −8.668640693757695192909685536691, −8.209318695485616783360084769529, −7.55069717245020761795585796219, −6.67556027171040608879739280615, −6.62487570409707098894129193587, −5.74182307183453133332150239171, −5.09333489016424149289860403188, −4.77850817902265293733670289695, −3.82573831139942949882045473030, −3.28655574158310369613010943963, −2.69683568595928777468641275575, −1.15156768478114842076101098008, −0.952698738893461562569621496680,
0.952698738893461562569621496680, 1.15156768478114842076101098008, 2.69683568595928777468641275575, 3.28655574158310369613010943963, 3.82573831139942949882045473030, 4.77850817902265293733670289695, 5.09333489016424149289860403188, 5.74182307183453133332150239171, 6.62487570409707098894129193587, 6.67556027171040608879739280615, 7.55069717245020761795585796219, 8.209318695485616783360084769529, 8.668640693757695192909685536691, 8.788271752617548098974140432651, 9.638518538297378943700459105380, 9.975510360396630275980711510891, 10.63816751645279271955505185667, 10.96070596289974715157243391367, 11.37096372431685723576425127635, 12.05037237374991078673053131079
