| L(s) = 1 | + 2·4-s − 4·11-s + 16-s + 4·29-s − 8·44-s + 2·49-s − 2·64-s + 4·89-s + 4·101-s − 4·109-s + 8·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2·4-s − 4·11-s + 16-s + 4·29-s − 8·44-s + 2·49-s − 2·64-s + 4·89-s + 4·101-s − 4·109-s + 8·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.299116526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.299116526\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.10140174808387906013027934624, −7.05524818227409851211111838737, −6.53777495137254431539820682059, −6.44072101847433800061126078318, −6.31702041722680136318241762076, −6.05973392456649966214617019390, −5.99505801324284614398595788577, −5.36254197613859972415809907923, −5.26893729319071190053609499447, −5.26448056870679861337229345934, −5.00790031982944233323773759027, −4.63425848457320099501599444232, −4.54862831784278500547078260021, −4.22074369125357980311155451450, −3.89389614320585964282021669759, −3.44684535440597905282257815055, −3.07411036406840727734834348561, −2.97809433125043087362951389168, −2.71814805403322930822498927913, −2.56501445190796848792842037496, −2.38794566889002732551603272968, −2.03049900619869313705408462111, −1.92391361406174110137308273157, −1.16397874980916275943415103296, −0.74081788545939650731209124912,
0.74081788545939650731209124912, 1.16397874980916275943415103296, 1.92391361406174110137308273157, 2.03049900619869313705408462111, 2.38794566889002732551603272968, 2.56501445190796848792842037496, 2.71814805403322930822498927913, 2.97809433125043087362951389168, 3.07411036406840727734834348561, 3.44684535440597905282257815055, 3.89389614320585964282021669759, 4.22074369125357980311155451450, 4.54862831784278500547078260021, 4.63425848457320099501599444232, 5.00790031982944233323773759027, 5.26448056870679861337229345934, 5.26893729319071190053609499447, 5.36254197613859972415809907923, 5.99505801324284614398595788577, 6.05973392456649966214617019390, 6.31702041722680136318241762076, 6.44072101847433800061126078318, 6.53777495137254431539820682059, 7.05524818227409851211111838737, 7.10140174808387906013027934624
