| L(s) = 1 | − 2-s + 3-s + 2·4-s + 4·5-s − 6-s + 4·7-s − 5·8-s − 4·10-s − 4·11-s + 2·12-s − 4·14-s + 4·15-s + 5·16-s − 2·17-s + 8·20-s + 4·21-s + 4·22-s − 5·24-s + 2·25-s − 27-s + 8·28-s + 10·29-s − 4·30-s + 8·31-s − 10·32-s − 4·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 4-s + 1.78·5-s − 0.408·6-s + 1.51·7-s − 1.76·8-s − 1.26·10-s − 1.20·11-s + 0.577·12-s − 1.06·14-s + 1.03·15-s + 5/4·16-s − 0.485·17-s + 1.78·20-s + 0.872·21-s + 0.852·22-s − 1.02·24-s + 2/5·25-s − 0.192·27-s + 1.51·28-s + 1.85·29-s − 0.730·30-s + 1.43·31-s − 1.76·32-s − 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.523370347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.523370347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−11.35278456111505393140591701449, −10.52099177368355250117715276635, −10.20407440138458392178190964046, −9.860223230300309532973443219191, −9.229718501172201749401543322655, −9.057260385329864963195085802075, −8.304382427255966794059718836807, −8.217346709228614328483705371552, −7.74811839956405163361973355805, −7.04714025670043888614968497425, −6.54800514714370983224343240132, −5.91110512932443020117134210164, −5.85732787959976694710321777273, −5.01380420270862060626962663481, −4.73897667355618378839530485033, −3.65206607630479313729802300122, −2.69806799877585015634573298336, −2.38957587810004361897357873419, −2.09495784516660473180817309851, −1.07960758519514121756710090243,
1.07960758519514121756710090243, 2.09495784516660473180817309851, 2.38957587810004361897357873419, 2.69806799877585015634573298336, 3.65206607630479313729802300122, 4.73897667355618378839530485033, 5.01380420270862060626962663481, 5.85732787959976694710321777273, 5.91110512932443020117134210164, 6.54800514714370983224343240132, 7.04714025670043888614968497425, 7.74811839956405163361973355805, 8.217346709228614328483705371552, 8.304382427255966794059718836807, 9.057260385329864963195085802075, 9.229718501172201749401543322655, 9.860223230300309532973443219191, 10.20407440138458392178190964046, 10.52099177368355250117715276635, 11.35278456111505393140591701449
