| L(s) = 1 | − 5·7-s + 2·19-s + 10·25-s − 8·31-s + 2·37-s + 18·49-s − 14·103-s + 4·109-s + 22·121-s + 127-s + 131-s − 10·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s − 50·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 0.458·19-s + 2·25-s − 1.43·31-s + 0.328·37-s + 18/7·49-s − 1.37·103-s + 0.383·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s − 0.867·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + 0.0760·173-s − 3.77·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.404411611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404411611\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−8.940862497842335180257010594715, −8.730946668540202363779368830964, −8.196928458065173802739480179206, −7.68864693948181262542318317339, −7.31236684886691527185828898520, −7.00087654990579005132539029906, −6.63235754814692409934266158721, −6.40583952799669887976218667675, −5.85617729892471095589469647032, −5.54080668261508826367836218049, −5.15488220060993446885474017736, −4.63284382844607820728718813521, −4.11460996600220646220158461440, −3.71530613657385921574732772242, −3.19200314612979602348259899630, −2.99707027982598922428588140467, −2.50104724660868707014546709584, −1.83324001589955420355968465519, −1.04762214568876821653473194168, −0.42674313281574924849884234814,
0.42674313281574924849884234814, 1.04762214568876821653473194168, 1.83324001589955420355968465519, 2.50104724660868707014546709584, 2.99707027982598922428588140467, 3.19200314612979602348259899630, 3.71530613657385921574732772242, 4.11460996600220646220158461440, 4.63284382844607820728718813521, 5.15488220060993446885474017736, 5.54080668261508826367836218049, 5.85617729892471095589469647032, 6.40583952799669887976218667675, 6.63235754814692409934266158721, 7.00087654990579005132539029906, 7.31236684886691527185828898520, 7.68864693948181262542318317339, 8.196928458065173802739480179206, 8.730946668540202363779368830964, 8.940862497842335180257010594715
