| L(s) = 1 | − 2-s + 2·3-s + 4-s − 4·5-s − 2·6-s − 3·8-s + 3·9-s + 4·10-s + 9·11-s + 2·12-s − 2·13-s − 8·15-s + 16-s − 5·17-s − 3·18-s − 6·19-s − 4·20-s − 9·22-s − 6·24-s + 2·25-s + 2·26-s + 4·27-s − 29-s + 8·30-s − 15·31-s + 32-s + 18·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 1.78·5-s − 0.816·6-s − 1.06·8-s + 9-s + 1.26·10-s + 2.71·11-s + 0.577·12-s − 0.554·13-s − 2.06·15-s + 1/4·16-s − 1.21·17-s − 0.707·18-s − 1.37·19-s − 0.894·20-s − 1.91·22-s − 1.22·24-s + 2/5·25-s + 0.392·26-s + 0.769·27-s − 0.185·29-s + 1.46·30-s − 2.69·31-s + 0.176·32-s + 3.13·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.405091065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.405091065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 15 T + 114 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 90 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 13 T + 160 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13 T + 146 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 60 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 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
−8.169639455523469205995892284691, −8.131928472170879572856800513885, −7.54082604004071313771553880702, −7.22073987252636197244708812392, −7.07156664385614049878229263333, −6.49268627126598742717016104058, −6.46065620582856289235945884890, −6.08709395525927400775595693764, −5.31470508436499007594936147498, −4.88670174878828617795899535912, −4.17773838342022286915195821725, −4.14639898833384675754932828630, −3.83900963069345298776932357639, −3.50064004839999450676931208595, −3.22128901745501234479983521587, −2.27397502199768002540044468174, −2.09845582373673996782577839472, −1.85380477883707391145628812344, −0.852704754571029603161369632427, −0.39263781656245326990218885502,
0.39263781656245326990218885502, 0.852704754571029603161369632427, 1.85380477883707391145628812344, 2.09845582373673996782577839472, 2.27397502199768002540044468174, 3.22128901745501234479983521587, 3.50064004839999450676931208595, 3.83900963069345298776932357639, 4.14639898833384675754932828630, 4.17773838342022286915195821725, 4.88670174878828617795899535912, 5.31470508436499007594936147498, 6.08709395525927400775595693764, 6.46065620582856289235945884890, 6.49268627126598742717016104058, 7.07156664385614049878229263333, 7.22073987252636197244708812392, 7.54082604004071313771553880702, 8.131928472170879572856800513885, 8.169639455523469205995892284691
