| L(s) = 1 | + 2·3-s + 3·9-s + 4·13-s − 4·16-s − 4·17-s − 2·23-s + 25-s + 4·27-s + 10·29-s + 8·39-s + 18·43-s − 8·48-s − 49-s − 8·51-s + 18·53-s − 16·61-s − 4·69-s + 2·75-s + 30·79-s + 5·81-s + 20·87-s − 16·101-s − 32·103-s − 24·107-s − 42·113-s + 12·117-s + 22·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.10·13-s − 16-s − 0.970·17-s − 0.417·23-s + 1/5·25-s + 0.769·27-s + 1.85·29-s + 1.28·39-s + 2.74·43-s − 1.15·48-s − 1/7·49-s − 1.12·51-s + 2.47·53-s − 2.04·61-s − 0.481·69-s + 0.230·75-s + 3.37·79-s + 5/9·81-s + 2.14·87-s − 1.59·101-s − 3.15·103-s − 2.32·107-s − 3.95·113-s + 1.10·117-s + 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.247117341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.247117341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−12.53711924879391049248068412288, −11.75942999023472771407128015184, −11.00019349193892584344335973713, −10.65135977649544478028872182113, −10.44237773289844884506708290251, −9.495803727925246431264251006851, −9.091325888207403476248922949636, −9.012452303660284311020256051581, −8.132154481969938936747522698136, −8.106523840292540468554590842384, −7.29523439370080668495912550963, −6.60369478835799314994114036829, −6.47642883402419071704485276541, −5.58698705029836725130809403607, −4.81595767110589909730428898467, −4.00210781752750486635416871863, −3.98036133172615340916797878280, −2.65376140931775816657558005927, −2.51396560601296405833713333615, −1.25303551678348609773086333816,
1.25303551678348609773086333816, 2.51396560601296405833713333615, 2.65376140931775816657558005927, 3.98036133172615340916797878280, 4.00210781752750486635416871863, 4.81595767110589909730428898467, 5.58698705029836725130809403607, 6.47642883402419071704485276541, 6.60369478835799314994114036829, 7.29523439370080668495912550963, 8.106523840292540468554590842384, 8.132154481969938936747522698136, 9.012452303660284311020256051581, 9.091325888207403476248922949636, 9.495803727925246431264251006851, 10.44237773289844884506708290251, 10.65135977649544478028872182113, 11.00019349193892584344335973713, 11.75942999023472771407128015184, 12.53711924879391049248068412288
