| L(s) = 1 | − 4·2-s + 6·4-s + 2·5-s − 4·8-s + 4·9-s − 8·10-s − 4·11-s + 14·13-s + 3·16-s + 12·17-s − 16·18-s + 12·19-s + 12·20-s + 16·22-s + 3·25-s − 56·26-s − 14·29-s + 8·31-s − 48·34-s + 24·36-s − 4·37-s − 48·38-s − 8·40-s + 6·41-s − 4·43-s − 24·44-s + 8·45-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s + 0.894·5-s − 1.41·8-s + 4/3·9-s − 2.52·10-s − 1.20·11-s + 3.88·13-s + 3/4·16-s + 2.91·17-s − 3.77·18-s + 2.75·19-s + 2.68·20-s + 3.41·22-s + 3/5·25-s − 10.9·26-s − 2.59·29-s + 1.43·31-s − 8.23·34-s + 4·36-s − 0.657·37-s − 7.78·38-s − 1.26·40-s + 0.937·41-s − 0.609·43-s − 3.61·44-s + 1.19·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143832218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143832218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 2 | $D_{4}$ | \( ( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2^3$ | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 14 T^{3} - 36 T^{4} + 14 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 14 T + 97 T^{2} + 574 T^{3} + 3276 T^{4} + 574 p T^{5} + 97 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T - 12 T^{2} - 112 T^{3} + 2831 T^{4} - 112 p T^{5} - 12 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 47 T^{2} - 6 T^{3} + 3732 T^{4} - 6 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 72 T^{2} + 8 T^{3} + 5207 T^{4} + 8 p T^{5} - 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 50 T^{2} + 112 T^{3} + 1395 T^{4} + 112 p T^{5} - 50 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 136 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14 T + 33 T^{2} - 574 T^{3} + 10892 T^{4} - 574 p T^{5} + 33 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 116 T^{2} + 8967 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T - 2 T^{2} - 48 T^{3} + 6051 T^{4} - 48 p T^{5} - 2 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 10 T - 39 T^{2} - 70 T^{3} + 6692 T^{4} - 70 p T^{5} - 39 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 12 T - 32 T^{2} + 216 T^{3} + 13359 T^{4} + 216 p T^{5} - 32 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 16 T + 6 T^{2} + 896 T^{3} + 28259 T^{4} + 896 p T^{5} + 6 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.77251768708008196263479562709, −7.58572877656281436645469969708, −7.32472395284869883511343560244, −7.07346795670219371233675574604, −7.01178059864762137243654610675, −6.44438730854059947446506890014, −6.09538827223564718170453980482, −6.07596108670925625862274174616, −5.73264628236579170618527344802, −5.61063202895794141311040928795, −5.31508618830392141361147148249, −5.18026382451802604854416625686, −4.86366888583629807037465188365, −4.25306176559966417966256741878, −3.89486135117042234384179969046, −3.80230569223751793404263871026, −3.28628843028964477457574637428, −3.27367253939449758510185649677, −3.12648490201576576722665127460, −2.21455674358685738671728136838, −2.11896338093253648734131547997, −1.37914378363513786209924942100, −1.07613077015127637854158133368, −0.909420791150851291428313096137, −0.907469985273236493169623419300,
0.907469985273236493169623419300, 0.909420791150851291428313096137, 1.07613077015127637854158133368, 1.37914378363513786209924942100, 2.11896338093253648734131547997, 2.21455674358685738671728136838, 3.12648490201576576722665127460, 3.27367253939449758510185649677, 3.28628843028964477457574637428, 3.80230569223751793404263871026, 3.89486135117042234384179969046, 4.25306176559966417966256741878, 4.86366888583629807037465188365, 5.18026382451802604854416625686, 5.31508618830392141361147148249, 5.61063202895794141311040928795, 5.73264628236579170618527344802, 6.07596108670925625862274174616, 6.09538827223564718170453980482, 6.44438730854059947446506890014, 7.01178059864762137243654610675, 7.07346795670219371233675574604, 7.32472395284869883511343560244, 7.58572877656281436645469969708, 7.77251768708008196263479562709
