| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 4·5-s + 8·6-s + 8·9-s + 8·10-s − 6·11-s − 8·12-s + 16·15-s − 4·16-s − 12·17-s − 16·18-s − 8·19-s − 8·20-s + 12·22-s + 8·25-s − 12·27-s − 2·29-s − 32·30-s + 8·31-s + 8·32-s + 24·33-s + 24·34-s + 16·36-s + 6·37-s + 16·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s − 1.78·5-s + 3.26·6-s + 8/3·9-s + 2.52·10-s − 1.80·11-s − 2.30·12-s + 4.13·15-s − 16-s − 2.91·17-s − 3.77·18-s − 1.83·19-s − 1.78·20-s + 2.55·22-s + 8/5·25-s − 2.30·27-s − 0.371·29-s − 5.84·30-s + 1.43·31-s + 1.41·32-s + 4.17·33-s + 4.11·34-s + 8/3·36-s + 0.986·37-s + 2.59·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−13.27543726386632237023834736288, −12.77754211622764385398692808552, −11.84585336871961339655873918622, −11.53430190791402638111985601815, −11.12855939747455399766224879908, −11.03615526563313727971497697102, −10.20206174900752305733710802634, −10.17545173493160079326317871982, −8.894601843176605624801842086876, −8.306840649573648139154730327668, −8.126756490343487190475146306068, −7.20565240330284028488283304296, −6.73005728304274755582986553004, −6.31546946995081892233015420942, −5.27249457573811292648996628312, −4.48975487947431727450468077844, −4.32853888064598947732097655777, −2.37699726942469351811773954673, 0, 0,
2.37699726942469351811773954673, 4.32853888064598947732097655777, 4.48975487947431727450468077844, 5.27249457573811292648996628312, 6.31546946995081892233015420942, 6.73005728304274755582986553004, 7.20565240330284028488283304296, 8.126756490343487190475146306068, 8.306840649573648139154730327668, 8.894601843176605624801842086876, 10.17545173493160079326317871982, 10.20206174900752305733710802634, 11.03615526563313727971497697102, 11.12855939747455399766224879908, 11.53430190791402638111985601815, 11.84585336871961339655873918622, 12.77754211622764385398692808552, 13.27543726386632237023834736288
