| L(s) = 1 | + 64·4-s − 243·9-s + 3.07e3·16-s − 6.25e3·25-s − 1.55e4·36-s + 1.33e4·37-s − 4.49e4·43-s + 1.31e5·64-s − 7.58e4·67-s + 1.81e5·79-s + 5.90e4·81-s − 4.00e5·100-s + 4.95e5·109-s + 3.22e5·121-s + 127-s + 131-s + 137-s + 139-s − 7.46e5·144-s + 8.52e5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 5.60e5·169-s − 2.87e6·172-s + ⋯ |
| L(s) = 1 | + 2·4-s − 9-s + 3·16-s − 2·25-s − 2·36-s + 1.59·37-s − 3.70·43-s + 4·64-s − 2.06·67-s + 3.27·79-s + 81-s − 4·100-s + 3.99·109-s + 2·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 3·144-s + 3.19·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 1.50·169-s − 7.41·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.790590819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.790590819\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 427 T + p^{5} T^{2} )( 1 + 427 T + p^{5} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 3143 T + p^{5} T^{2} )( 1 + 3143 T + p^{5} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2723 T + p^{5} T^{2} )( 1 + 2723 T + p^{5} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6661 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 22475 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 38626 T + p^{5} T^{2} )( 1 + 38626 T + p^{5} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 37939 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 78127 T + p^{5} T^{2} )( 1 + 78127 T + p^{5} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 90857 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 134386 T + p^{5} T^{2} )( 1 + 134386 T + p^{5} T^{2} ) \) |
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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
−12.05209667589065974776508992130, −11.80383264807515841549356492128, −11.49734065026930261320698469912, −11.00677069816009224198216718914, −10.52176348900673877468694182112, −9.871762248068828598134417346377, −9.568641317143513811998128737777, −8.487735949416690918645448220058, −8.128821895189613885053504448728, −7.61818635230342723819810588003, −7.05305830452699616447645318448, −6.36362963533110108365532922463, −6.02292028923262869382215871363, −5.53488486678782408201045875137, −4.61818483850676143900332679746, −3.33781404314844089602731556852, −3.24023912076418603559457399214, −2.10094004331203923642477091921, −1.86243862887864202761484032072, −0.61319383602466123003363729961,
0.61319383602466123003363729961, 1.86243862887864202761484032072, 2.10094004331203923642477091921, 3.24023912076418603559457399214, 3.33781404314844089602731556852, 4.61818483850676143900332679746, 5.53488486678782408201045875137, 6.02292028923262869382215871363, 6.36362963533110108365532922463, 7.05305830452699616447645318448, 7.61818635230342723819810588003, 8.128821895189613885053504448728, 8.487735949416690918645448220058, 9.568641317143513811998128737777, 9.871762248068828598134417346377, 10.52176348900673877468694182112, 11.00677069816009224198216718914, 11.49734065026930261320698469912, 11.80383264807515841549356492128, 12.05209667589065974776508992130
