| L(s) = 1 | − 2·2-s + 2·4-s − 2·7-s + 4·14-s − 4·16-s + 2·17-s − 10·23-s + 6·25-s − 4·28-s − 14·31-s + 8·32-s − 4·34-s + 4·41-s + 20·46-s + 3·49-s − 12·50-s + 28·62-s − 8·64-s + 4·68-s − 30·71-s − 28·79-s − 8·82-s − 26·89-s − 20·92-s − 28·97-s − 6·98-s + 12·100-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.755·7-s + 1.06·14-s − 16-s + 0.485·17-s − 2.08·23-s + 6/5·25-s − 0.755·28-s − 2.51·31-s + 1.41·32-s − 0.685·34-s + 0.624·41-s + 2.94·46-s + 3/7·49-s − 1.69·50-s + 3.55·62-s − 64-s + 0.485·68-s − 3.56·71-s − 3.15·79-s − 0.883·82-s − 2.75·89-s − 2.08·92-s − 2.84·97-s − 0.606·98-s + 6/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2122400902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2122400902\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.858836300135979305976447501723, −9.192392812184509574549061191963, −8.935168772701952214420735604333, −8.547536848463692056793550593836, −8.239869611956688114807726572980, −7.57497846125810977950257510947, −7.46850419789980652659799474827, −6.96582062291966178022602929668, −6.71349829997619331367582564672, −5.93579477733754355163933939893, −5.74872591673051753796580018429, −5.37029345297608884360089259703, −4.33117788465248853942571840832, −4.32943916376496872714672702190, −3.65446722905062973768644923531, −2.93910021077498731887237177472, −2.60911414813695091113651228143, −1.67577352415054073113134653995, −1.43751376113978214378912348920, −0.23429795009191015451726674852,
0.23429795009191015451726674852, 1.43751376113978214378912348920, 1.67577352415054073113134653995, 2.60911414813695091113651228143, 2.93910021077498731887237177472, 3.65446722905062973768644923531, 4.32943916376496872714672702190, 4.33117788465248853942571840832, 5.37029345297608884360089259703, 5.74872591673051753796580018429, 5.93579477733754355163933939893, 6.71349829997619331367582564672, 6.96582062291966178022602929668, 7.46850419789980652659799474827, 7.57497846125810977950257510947, 8.239869611956688114807726572980, 8.547536848463692056793550593836, 8.935168772701952214420735604333, 9.192392812184509574549061191963, 9.858836300135979305976447501723
