| L(s) = 1 | − 5-s + 7-s − 6·11-s + 13-s − 2·19-s − 6·23-s − 6·29-s − 8·31-s − 35-s − 14·37-s + 6·41-s + 4·43-s − 12·47-s + 7·49-s − 12·53-s + 6·55-s − 11·61-s − 65-s + 7·67-s − 12·71-s + 22·73-s − 6·77-s + 79-s − 6·83-s − 24·89-s + 91-s + 2·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.80·11-s + 0.277·13-s − 0.458·19-s − 1.25·23-s − 1.11·29-s − 1.43·31-s − 0.169·35-s − 2.30·37-s + 0.937·41-s + 0.609·43-s − 1.75·47-s + 49-s − 1.64·53-s + 0.809·55-s − 1.40·61-s − 0.124·65-s + 0.855·67-s − 1.42·71-s + 2.57·73-s − 0.683·77-s + 0.112·79-s − 0.658·83-s − 2.54·89-s + 0.104·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2237350495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2237350495\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.684165220106962423030180721426, −8.961076822919021374393900967243, −8.949883376218103070743808538599, −8.243469341778876319583538226088, −7.949043532999002027245448905818, −7.66239061326064564193665794442, −7.47101832687215880270867037410, −6.63058003134488918092079384721, −6.59004112898116291910758319679, −5.70196077856203051079475351937, −5.50128365907402003025237188560, −5.19800022816892829605052492417, −4.62823457667509550779019639423, −4.04850110351683053181922841006, −3.74908654342524861089860561508, −3.13249868820268966518145681601, −2.62541077174039872468217678732, −1.91731612969935062924872538576, −1.60095539050902192641329435932, −0.17246342800236113239458994900,
0.17246342800236113239458994900, 1.60095539050902192641329435932, 1.91731612969935062924872538576, 2.62541077174039872468217678732, 3.13249868820268966518145681601, 3.74908654342524861089860561508, 4.04850110351683053181922841006, 4.62823457667509550779019639423, 5.19800022816892829605052492417, 5.50128365907402003025237188560, 5.70196077856203051079475351937, 6.59004112898116291910758319679, 6.63058003134488918092079384721, 7.47101832687215880270867037410, 7.66239061326064564193665794442, 7.949043532999002027245448905818, 8.243469341778876319583538226088, 8.949883376218103070743808538599, 8.961076822919021374393900967243, 9.684165220106962423030180721426
