| L(s) = 1 | − 4·2-s − 4·3-s + 8·4-s − 8·5-s + 16·6-s − 8·8-s + 4·9-s + 32·10-s − 12·11-s − 32·12-s + 32·15-s − 4·16-s − 16·18-s − 4·19-s − 64·20-s + 48·22-s − 4·23-s + 32·24-s + 40·25-s + 4·27-s − 16·29-s − 128·30-s − 24·31-s + 32·32-s + 48·33-s + 32·36-s − 16·37-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 4·4-s − 3.57·5-s + 6.53·6-s − 2.82·8-s + 4/3·9-s + 10.1·10-s − 3.61·11-s − 9.23·12-s + 8.26·15-s − 16-s − 3.77·18-s − 0.917·19-s − 14.3·20-s + 10.2·22-s − 0.834·23-s + 6.53·24-s + 8·25-s + 0.769·27-s − 2.97·29-s − 23.3·30-s − 4.31·31-s + 5.65·32-s + 8.35·33-s + 16/3·36-s − 2.63·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 4 T + 4 p T^{2} + 28 T^{3} + 56 T^{4} + 28 p T^{5} + 4 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 8 T + 24 T^{2} + 32 T^{3} + 32 T^{4} + 32 p T^{5} + 24 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12 T + 86 T^{2} + 428 T^{3} + 1634 T^{4} + 428 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 72 T^{3} + 32 T^{4} + 72 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T + 36 T^{2} + 196 T^{3} + 920 T^{4} + 196 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 16 T + 162 T^{2} + 1216 T^{3} + 7490 T^{4} + 1216 p T^{5} + 162 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 16 T + 162 T^{2} + 1312 T^{3} + 9026 T^{4} + 1312 p T^{5} + 162 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 360 T^{3} + 4034 T^{4} + 360 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T + 114 T^{2} + 632 T^{3} + 3810 T^{4} + 632 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5766 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 20 T + 118 T^{2} + 412 T^{3} - 8478 T^{4} + 412 p T^{5} + 118 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 20 T + 300 T^{2} + 3180 T^{3} + 28600 T^{4} + 3180 p T^{5} + 300 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 16 T + 96 T^{2} + 256 T^{3} + 512 T^{4} + 256 p T^{5} + 96 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 226 T^{2} - 2456 T^{3} + 23362 T^{4} - 2456 p T^{5} + 226 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 1154 T^{4} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 3096 T^{3} + 30146 T^{4} - 3096 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 132 T^{2} + 1236 T^{3} + 11608 T^{4} + 1236 p T^{5} + 132 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 6432 T^{3} + 68258 T^{4} + 6432 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 202 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.476213480338880563508409335678, −8.957861858034234570810244812229, −8.846134160901408277104581993178, −8.773808095506330094418033374357, −8.203221857541121846376713502020, −8.153882298323288257339066305161, −7.83958791277833236794267892183, −7.81945012662596335823302185270, −7.66736408639700318581125906583, −7.06858680240611138633294870478, −7.00839530588096599093699724654, −6.97711291655205444164352179646, −6.64860267540627262234622253763, −5.74970298381492036367853470277, −5.65424998802923106598365758339, −5.25860399477540022808461512775, −5.23259133295239489655345610239, −4.94061853535647076738824538356, −4.52918745644796550103106267128, −3.81927197185510651802505405975, −3.79388063117807347582304027346, −3.40735467730031703534106433827, −2.82810455494575849020957114256, −2.09573197158845129667852700435, −1.73567312311397554560939721570, 0, 0, 0, 0,
1.73567312311397554560939721570, 2.09573197158845129667852700435, 2.82810455494575849020957114256, 3.40735467730031703534106433827, 3.79388063117807347582304027346, 3.81927197185510651802505405975, 4.52918745644796550103106267128, 4.94061853535647076738824538356, 5.23259133295239489655345610239, 5.25860399477540022808461512775, 5.65424998802923106598365758339, 5.74970298381492036367853470277, 6.64860267540627262234622253763, 6.97711291655205444164352179646, 7.00839530588096599093699724654, 7.06858680240611138633294870478, 7.66736408639700318581125906583, 7.81945012662596335823302185270, 7.83958791277833236794267892183, 8.153882298323288257339066305161, 8.203221857541121846376713502020, 8.773808095506330094418033374357, 8.846134160901408277104581993178, 8.957861858034234570810244812229, 9.476213480338880563508409335678
