L(s) = 1 | + 16·4-s + 70·13-s + 192·16-s + 250·25-s − 1.04e3·43-s − 286·49-s + 1.12e3·52-s − 364·61-s + 2.04e3·64-s − 1.76e3·79-s + 4.00e3·100-s + 3.64e3·103-s + 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.70e3·169-s − 1.66e4·172-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 2·4-s + 1.49·13-s + 3·16-s + 2·25-s − 3.68·43-s − 0.833·49-s + 2.98·52-s − 0.764·61-s + 4·64-s − 2.51·79-s + 4·100-s + 3.48·103-s + 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.23·169-s − 7.37·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.992252686\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.992252686\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 70 T + p^{3} T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )( 1 + 20 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )( 1 + 56 T + p^{3} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 110 T + p^{3} T^{2} )( 1 + 110 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 520 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 880 T + p^{3} T^{2} )( 1 + 880 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )( 1 + 1190 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 884 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )( 1 + 1330 T + p^{3} T^{2} ) \) |
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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
−13.09793682249852029627757446505, −12.81104306052950314221631167958, −12.15207452153091951168640441358, −11.60737451788467510518812776262, −11.22253296543019460382726301899, −10.92732715173298755552201689528, −10.13361477787635338604374478398, −10.05881747090315228572785022402, −8.767159425908710069810667880785, −8.500916034879971704384469377688, −7.80401112502456303824670417537, −7.07854376181715442610251502412, −6.65201975302685759069487382070, −6.22795978536921027379837434826, −5.54771547783922769648659440040, −4.68619552344503991237305000032, −3.28794952208295891947377017488, −3.19344125700380133116637973287, −1.92033501367808232448959916594, −1.20959907316659183120245590820,
1.20959907316659183120245590820, 1.92033501367808232448959916594, 3.19344125700380133116637973287, 3.28794952208295891947377017488, 4.68619552344503991237305000032, 5.54771547783922769648659440040, 6.22795978536921027379837434826, 6.65201975302685759069487382070, 7.07854376181715442610251502412, 7.80401112502456303824670417537, 8.500916034879971704384469377688, 8.767159425908710069810667880785, 10.05881747090315228572785022402, 10.13361477787635338604374478398, 10.92732715173298755552201689528, 11.22253296543019460382726301899, 11.60737451788467510518812776262, 12.15207452153091951168640441358, 12.81104306052950314221631167958, 13.09793682249852029627757446505