| L(s) = 1 | − 2·4-s + 3·16-s − 8·19-s + 10·25-s + 16·29-s − 8·41-s − 4·49-s − 24·59-s + 16·61-s − 4·64-s + 48·71-s + 16·76-s − 32·79-s − 48·89-s − 20·100-s + 16·101-s + 16·109-s − 32·116-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4-s + 3/4·16-s − 1.83·19-s + 2·25-s + 2.97·29-s − 1.24·41-s − 4/7·49-s − 3.12·59-s + 2.04·61-s − 1/2·64-s + 5.69·71-s + 1.83·76-s − 3.60·79-s − 5.08·89-s − 2·100-s + 1.59·101-s + 1.53·109-s − 2.97·116-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04624049809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04624049809\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 22 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 29 | $C_4$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2902 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 698 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 5302 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_4$ | \( ( 1 - 24 T + 266 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 8038 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 22998 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 24 T + 302 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 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
−6.42143299297456984010859849764, −6.36154733452225212839352035500, −6.28155153711506909365717298586, −5.76175304007398279159461323534, −5.56311272426857190537098888549, −5.33831870441393808485397365356, −5.28589147782287229547898173779, −4.91283720039243270787854640997, −4.62474942079992581014282431392, −4.58593404830198823505728042798, −4.52107986697149628280879841610, −4.14132704661120031264026344449, −3.97839368542850525334343712531, −3.60515532911551627500492020694, −3.44392608435714871830441050135, −3.14038404819300011738880556718, −2.80526997437789417724925754802, −2.75435518301969108419080559644, −2.44627752833604455209522163261, −2.07400847565888033570668238387, −1.79640580063365056967993841645, −1.19316608095808320386598525250, −1.18720858131755513538249794808, −0.813083232012666277552065544936, −0.04264960859566439663942372292,
0.04264960859566439663942372292, 0.813083232012666277552065544936, 1.18720858131755513538249794808, 1.19316608095808320386598525250, 1.79640580063365056967993841645, 2.07400847565888033570668238387, 2.44627752833604455209522163261, 2.75435518301969108419080559644, 2.80526997437789417724925754802, 3.14038404819300011738880556718, 3.44392608435714871830441050135, 3.60515532911551627500492020694, 3.97839368542850525334343712531, 4.14132704661120031264026344449, 4.52107986697149628280879841610, 4.58593404830198823505728042798, 4.62474942079992581014282431392, 4.91283720039243270787854640997, 5.28589147782287229547898173779, 5.33831870441393808485397365356, 5.56311272426857190537098888549, 5.76175304007398279159461323534, 6.28155153711506909365717298586, 6.36154733452225212839352035500, 6.42143299297456984010859849764
