| L(s) = 1 | + 2-s − 3-s − 6-s − 3·7-s − 8-s − 11-s − 5·13-s − 3·14-s − 16-s + 5·19-s + 3·21-s − 22-s − 4·23-s + 24-s − 5·26-s + 27-s + 20·31-s + 33-s − 37-s + 5·38-s + 5·39-s − 6·41-s + 3·42-s − 2·43-s − 4·46-s + 18·47-s + 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 1.13·7-s − 0.353·8-s − 0.301·11-s − 1.38·13-s − 0.801·14-s − 1/4·16-s + 1.14·19-s + 0.654·21-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 0.980·26-s + 0.192·27-s + 3.59·31-s + 0.174·33-s − 0.164·37-s + 0.811·38-s + 0.800·39-s − 0.937·41-s + 0.462·42-s − 0.304·43-s − 0.589·46-s + 2.62·47-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.608392849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.608392849\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 47 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.726978647736411471558143350450, −8.890022516944410829498740888918, −8.654381527439837630128320362046, −8.283819276833153961063209403466, −7.58627358300409865246342951881, −7.39922300883911381168368945777, −6.82509599648557609154463421100, −6.70448904267932751802521540686, −5.93814285244655286523974198812, −5.89394001334550543821056620828, −5.34318906936787484282377759278, −5.02210567050117806721860115984, −4.43772612404413981603357721295, −4.18664178339003414703876735657, −3.59184455343915569332974804597, −3.03922883577930378766682149778, −2.47048326066602158493058280731, −2.43849390788336608312612640990, −1.07116076019876602016018627223, −0.49289627327543399518612616493,
0.49289627327543399518612616493, 1.07116076019876602016018627223, 2.43849390788336608312612640990, 2.47048326066602158493058280731, 3.03922883577930378766682149778, 3.59184455343915569332974804597, 4.18664178339003414703876735657, 4.43772612404413981603357721295, 5.02210567050117806721860115984, 5.34318906936787484282377759278, 5.89394001334550543821056620828, 5.93814285244655286523974198812, 6.70448904267932751802521540686, 6.82509599648557609154463421100, 7.39922300883911381168368945777, 7.58627358300409865246342951881, 8.283819276833153961063209403466, 8.654381527439837630128320362046, 8.890022516944410829498740888918, 9.726978647736411471558143350450
