| L(s) = 1 | + 2·11-s − 16·13-s − 36·23-s − 177·25-s − 292·37-s − 212·47-s − 613·49-s + 40·59-s − 816·61-s + 40·71-s − 1.18e3·73-s + 690·83-s − 2.48e3·97-s + 2.60e3·107-s − 2.85e3·109-s − 2.65e3·121-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.20e3·169-s + ⋯ |
| L(s) = 1 | + 0.0548·11-s − 0.341·13-s − 0.326·23-s − 1.41·25-s − 1.29·37-s − 0.657·47-s − 1.78·49-s + 0.0882·59-s − 1.71·61-s + 0.0668·71-s − 1.89·73-s + 0.912·83-s − 2.59·97-s + 2.35·107-s − 2.50·109-s − 1.99·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.0187·143-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.91·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 177 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 613 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 686 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 590 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 34470 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50749 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 146 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 130542 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 109666 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 106 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 208329 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 408 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 243826 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 591 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 518878 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 345 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1360590 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1241 T + p^{3} T^{2} )^{2} \) |
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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
−9.554815300127722072645067204089, −9.266070861932843597262957914338, −8.656738995869936030043444800282, −8.349191527719148576517431087599, −7.68267692219415489894855892301, −7.68189554116178492552370692166, −6.91596846982891880853375578760, −6.61881289405150127501994756630, −5.95204096341922468482861993086, −5.79029145471358520660590132511, −4.92025810508649731884298255518, −4.85482622930540250257261524227, −4.02540859805861371546374738476, −3.67304700046114069377336956258, −3.04093633715510623144612866085, −2.47787841802592656562575874995, −1.71508064180919529931453162425, −1.35054407013558576294512350135, 0, 0,
1.35054407013558576294512350135, 1.71508064180919529931453162425, 2.47787841802592656562575874995, 3.04093633715510623144612866085, 3.67304700046114069377336956258, 4.02540859805861371546374738476, 4.85482622930540250257261524227, 4.92025810508649731884298255518, 5.79029145471358520660590132511, 5.95204096341922468482861993086, 6.61881289405150127501994756630, 6.91596846982891880853375578760, 7.68189554116178492552370692166, 7.68267692219415489894855892301, 8.349191527719148576517431087599, 8.656738995869936030043444800282, 9.266070861932843597262957914338, 9.554815300127722072645067204089
