| L(s) = 1 | − 2·2-s + 2·4-s + 6·7-s − 24·11-s + 24·13-s − 12·14-s − 4·16-s + 24·17-s + 48·22-s + 6·23-s − 48·26-s + 12·28-s − 16·31-s + 8·32-s − 48·34-s + 96·37-s + 96·41-s + 54·43-s − 48·44-s − 12·46-s + 54·47-s + 18·49-s + 48·52-s − 24·53-s + 64·61-s + 32·62-s − 8·64-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s + 6/7·7-s − 2.18·11-s + 1.84·13-s − 6/7·14-s − 1/4·16-s + 1.41·17-s + 2.18·22-s + 6/23·23-s − 1.84·26-s + 3/7·28-s − 0.516·31-s + 1/4·32-s − 1.41·34-s + 2.59·37-s + 2.34·41-s + 1.25·43-s − 1.09·44-s − 0.260·46-s + 1.14·47-s + 0.367·49-s + 0.923·52-s − 0.452·53-s + 1.04·61-s + 0.516·62-s − 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.710069873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710069873\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 782 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 96 T + 4608 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 54 T + 1458 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T + 1458 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3362 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10882 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 186 T + 17298 T^{2} - 186 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14942 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.98816700885998946765037781109, −10.65070665597727047068646486991, −10.32077341497812238302166671179, −9.763698402816044382839187809700, −9.081880756409868057477351844910, −9.061516409740485338344520988386, −8.136274755116591802529441643843, −7.946680033049022121071910728037, −7.71580436606128620507935154608, −7.37623227060207952863675107331, −6.24793164507256641198846913758, −6.09959465441499361278566382897, −5.24063989221777173485652052904, −5.21717934895737897723979751904, −4.14065018367742846364065204204, −3.76251780683402980053370759450, −2.64197868092256633405935035009, −2.44953680416428328450775306204, −1.19029877160003910635911488839, −0.77792245781875828776542141108,
0.77792245781875828776542141108, 1.19029877160003910635911488839, 2.44953680416428328450775306204, 2.64197868092256633405935035009, 3.76251780683402980053370759450, 4.14065018367742846364065204204, 5.21717934895737897723979751904, 5.24063989221777173485652052904, 6.09959465441499361278566382897, 6.24793164507256641198846913758, 7.37623227060207952863675107331, 7.71580436606128620507935154608, 7.946680033049022121071910728037, 8.136274755116591802529441643843, 9.061516409740485338344520988386, 9.081880756409868057477351844910, 9.763698402816044382839187809700, 10.32077341497812238302166671179, 10.65070665597727047068646486991, 10.98816700885998946765037781109
