| L(s) = 1 | + 2-s − 7-s − 8-s − 5·13-s − 14-s − 16-s − 6·17-s + 4·19-s − 9·23-s + 5·25-s − 5·26-s − 3·29-s − 5·31-s − 6·34-s + 4·37-s + 4·38-s − 6·41-s + 43-s − 9·46-s − 6·47-s + 5·50-s − 6·53-s + 56-s − 3·58-s − 3·59-s + 10·61-s − 5·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.377·7-s − 0.353·8-s − 1.38·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.87·23-s + 25-s − 0.980·26-s − 0.557·29-s − 0.898·31-s − 1.02·34-s + 0.657·37-s + 0.648·38-s − 0.937·41-s + 0.152·43-s − 1.32·46-s − 0.875·47-s + 0.707·50-s − 0.824·53-s + 0.133·56-s − 0.393·58-s − 0.390·59-s + 1.28·61-s − 0.635·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.170491327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170491327\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.885134133042836330935516795291, −9.668560844692927382484509656630, −9.391262284233501507210926728986, −8.696429661124892141416874886261, −8.477560232599063629792787379997, −7.951236135915009545633414434809, −7.41594919424181699244440397273, −6.98406452730295600993489160984, −6.81751324330959583993421152007, −6.00348404784966079694422444278, −5.92166960088236342353435483151, −5.05883848137187692748835115015, −5.04999033406061977695829306295, −4.19752837815257050096715567336, −4.15878802283952430183615936459, −3.25039431342898257412006554155, −2.97517000591496477531489828277, −2.18334898427611332614545841639, −1.79398034052069660191843039230, −0.39941090450697708471924845253,
0.39941090450697708471924845253, 1.79398034052069660191843039230, 2.18334898427611332614545841639, 2.97517000591496477531489828277, 3.25039431342898257412006554155, 4.15878802283952430183615936459, 4.19752837815257050096715567336, 5.04999033406061977695829306295, 5.05883848137187692748835115015, 5.92166960088236342353435483151, 6.00348404784966079694422444278, 6.81751324330959583993421152007, 6.98406452730295600993489160984, 7.41594919424181699244440397273, 7.951236135915009545633414434809, 8.477560232599063629792787379997, 8.696429661124892141416874886261, 9.391262284233501507210926728986, 9.668560844692927382484509656630, 9.885134133042836330935516795291
