| L(s) = 1 | + 2-s + 5-s + 2·7-s − 8-s + 10-s − 11-s − 2·13-s + 2·14-s − 16-s + 19-s − 22-s − 23-s − 2·26-s + 2·35-s + 37-s + 38-s − 40-s + 2·41-s − 46-s − 47-s + 3·49-s − 53-s − 55-s − 2·56-s + 2·59-s + 64-s − 2·65-s + ⋯ |
| L(s) = 1 | + 2-s + 5-s + 2·7-s − 8-s + 10-s − 11-s − 2·13-s + 2·14-s − 16-s + 19-s − 22-s − 23-s − 2·26-s + 2·35-s + 37-s + 38-s − 40-s + 2·41-s − 46-s − 47-s + 3·49-s − 53-s − 55-s − 2·56-s + 2·59-s + 64-s − 2·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.437247008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.437247008\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.461865109041524568028347669085, −9.009573886587004097076194906078, −8.251572839608973853626951332520, −8.179402031079579970020860391638, −7.76539118298374296846084543204, −7.40113234378831909711612688961, −7.11692493458671415712059453737, −6.37139887040370844540446229686, −6.03372188414248506532996682906, −5.46552772426230396254416443571, −5.40003157953855464434691096290, −4.86596004918300078878185529404, −4.85849633372844742579355715621, −4.07851150548218327490554577510, −4.02596492776142049016658132521, −2.82891471962978275584765359106, −2.82700128728165045926069615453, −2.02178447027439540955782500175, −2.00959894272386569722165383402, −0.886730212315747579153929022677,
0.886730212315747579153929022677, 2.00959894272386569722165383402, 2.02178447027439540955782500175, 2.82700128728165045926069615453, 2.82891471962978275584765359106, 4.02596492776142049016658132521, 4.07851150548218327490554577510, 4.85849633372844742579355715621, 4.86596004918300078878185529404, 5.40003157953855464434691096290, 5.46552772426230396254416443571, 6.03372188414248506532996682906, 6.37139887040370844540446229686, 7.11692493458671415712059453737, 7.40113234378831909711612688961, 7.76539118298374296846084543204, 8.179402031079579970020860391638, 8.251572839608973853626951332520, 9.009573886587004097076194906078, 9.461865109041524568028347669085
