| L(s) = 1 | + 6·3-s − 4·4-s + 27·9-s − 24·12-s + 16·16-s − 260·17-s + 150·25-s + 108·27-s − 36·29-s − 108·36-s − 152·43-s + 96·48-s + 622·49-s − 1.56e3·51-s + 764·53-s + 716·61-s − 64·64-s + 1.04e3·68-s + 900·75-s − 1.95e3·79-s + 405·81-s − 216·87-s − 600·100-s − 1.03e3·101-s − 224·103-s − 744·107-s − 432·108-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s − 3.70·17-s + 6/5·25-s + 0.769·27-s − 0.230·29-s − 1/2·36-s − 0.539·43-s + 0.288·48-s + 1.81·49-s − 4.28·51-s + 1.98·53-s + 1.50·61-s − 1/8·64-s + 1.85·68-s + 1.38·75-s − 2.77·79-s + 5/9·81-s − 0.266·87-s − 3/5·100-s − 1.02·101-s − 0.214·103-s − 0.672·107-s − 0.384·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.197488327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197488327\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )( 1 + 4 p T + p^{3} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 622 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1062 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 130 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13318 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 25726 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 p T + p^{3} T^{2} )( 1 + 12 p T + p^{3} T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 6798 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 76 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3342 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 382 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 195462 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 358 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 111526 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 156318 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 341330 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 976 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 127510 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1260942 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1448350 T^{2} + p^{6} T^{4} \) |
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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
−9.670128744536759041032034745284, −9.124794221872116073003463240844, −8.789616090104045527964856577049, −8.706375114149408590148870097491, −8.458817569423969188568435150628, −7.76612767480301943300680066515, −7.16165388409966698832172935135, −6.85062451753392867027536244855, −6.73999239465243409347204606476, −5.92104777030879332334984403212, −5.45676042879623974439533014159, −4.64622862781921420867562534243, −4.55362670802625689011220061591, −4.01603530283211187039009158866, −3.62735962932438679534243527039, −2.76813292916274978802455244177, −2.41494561031306130401015048528, −2.03829401380890338725458088501, −1.16923579954615080831175118171, −0.35420853440194243953923057116,
0.35420853440194243953923057116, 1.16923579954615080831175118171, 2.03829401380890338725458088501, 2.41494561031306130401015048528, 2.76813292916274978802455244177, 3.62735962932438679534243527039, 4.01603530283211187039009158866, 4.55362670802625689011220061591, 4.64622862781921420867562534243, 5.45676042879623974439533014159, 5.92104777030879332334984403212, 6.73999239465243409347204606476, 6.85062451753392867027536244855, 7.16165388409966698832172935135, 7.76612767480301943300680066515, 8.458817569423969188568435150628, 8.706375114149408590148870097491, 8.789616090104045527964856577049, 9.124794221872116073003463240844, 9.670128744536759041032034745284
