| L(s) = 1 | + 2-s − 3·3-s − 2·4-s − 3·6-s + 7-s − 3·8-s + 2·9-s + 6·12-s + 8·13-s + 14-s + 16-s + 17-s + 2·18-s − 3·21-s − 3·23-s + 9·24-s + 8·26-s + 6·27-s − 2·28-s + 5·29-s − 6·31-s + 2·32-s + 34-s − 4·36-s − 16·37-s − 24·39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 4-s − 1.22·6-s + 0.377·7-s − 1.06·8-s + 2/3·9-s + 1.73·12-s + 2.21·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 0.654·21-s − 0.625·23-s + 1.83·24-s + 1.56·26-s + 1.15·27-s − 0.377·28-s + 0.928·29-s − 1.07·31-s + 0.353·32-s + 0.171·34-s − 2/3·36-s − 2.63·37-s − 3.84·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 133 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 123 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 121 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 23 T + 277 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 27 T + 337 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 193 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 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
−8.503170903185344574057746711540, −8.379364782521708086565917091396, −7.983376927756932264559051822831, −6.99479341832816540563379150673, −6.96529891248506738592981939645, −6.53632962195382402464807851300, −5.91006250767308799935454744714, −5.82971335465278032089922732194, −5.33708798379076769664459973325, −5.32768607833226008271780614057, −4.68128857940897950933708924402, −4.41524533997720725352978701617, −3.82068199272082018296731766608, −3.53569641063221469554030809087, −3.20431880332447588781811443012, −2.33867041360714091304038885032, −1.34613775953838081068712223844, −1.28864669032727297114672054915, 0, 0,
1.28864669032727297114672054915, 1.34613775953838081068712223844, 2.33867041360714091304038885032, 3.20431880332447588781811443012, 3.53569641063221469554030809087, 3.82068199272082018296731766608, 4.41524533997720725352978701617, 4.68128857940897950933708924402, 5.32768607833226008271780614057, 5.33708798379076769664459973325, 5.82971335465278032089922732194, 5.91006250767308799935454744714, 6.53632962195382402464807851300, 6.96529891248506738592981939645, 6.99479341832816540563379150673, 7.983376927756932264559051822831, 8.379364782521708086565917091396, 8.503170903185344574057746711540
