| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s − 7-s + 8-s + 3·9-s + 10-s + 11-s − 4·13-s + 14-s − 15-s − 16-s + 4·17-s − 3·18-s − 8·19-s − 21-s − 22-s + 23-s + 24-s + 4·26-s + 8·27-s − 18·29-s + 30-s − 2·31-s + 33-s − 4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 9-s + 0.316·10-s + 0.301·11-s − 1.10·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.707·18-s − 1.83·19-s − 0.218·21-s − 0.213·22-s + 0.208·23-s + 0.204·24-s + 0.784·26-s + 1.53·27-s − 3.34·29-s + 0.182·30-s − 0.359·31-s + 0.174·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9135454482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9135454482\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−10.34467220385752236620268643014, −9.884532617772672622808100804309, −9.796901175459375810086544189691, −9.144044935327297816535205476709, −9.024359342556082646431855598059, −8.229014046706414779608633266062, −8.195962417746597634886969137870, −7.37476008552137113706092814820, −7.35843489079672239310535681171, −6.85561272595111622785322484682, −6.29146568399933514754157951822, −5.66335509184139915189627053206, −5.14755307109228775436791498114, −4.32237046064832077499027145061, −4.30670245491343268400615565288, −3.43637722233213279630311974613, −3.13083680426458924304764141616, −1.93139348144554047455017181509, −1.90650933069563832207401384740, −0.51827449414454704169774619761,
0.51827449414454704169774619761, 1.90650933069563832207401384740, 1.93139348144554047455017181509, 3.13083680426458924304764141616, 3.43637722233213279630311974613, 4.30670245491343268400615565288, 4.32237046064832077499027145061, 5.14755307109228775436791498114, 5.66335509184139915189627053206, 6.29146568399933514754157951822, 6.85561272595111622785322484682, 7.35843489079672239310535681171, 7.37476008552137113706092814820, 8.195962417746597634886969137870, 8.229014046706414779608633266062, 9.024359342556082646431855598059, 9.144044935327297816535205476709, 9.796901175459375810086544189691, 9.884532617772672622808100804309, 10.34467220385752236620268643014
