| L(s) = 1 | − 6·3-s − 2·4-s + 10·7-s + 21·9-s − 12·11-s + 12·12-s + 48·17-s − 42·19-s − 60·21-s − 24·23-s − 54·27-s − 20·28-s + 102·31-s + 72·33-s − 42·36-s − 22·37-s − 28·43-s + 24·44-s + 132·47-s + 49·49-s − 288·51-s − 120·53-s + 252·57-s − 24·59-s − 72·61-s + 210·63-s + 8·64-s + ⋯ |
| L(s) = 1 | − 2·3-s − 1/2·4-s + 10/7·7-s + 7/3·9-s − 1.09·11-s + 12-s + 2.82·17-s − 2.21·19-s − 2.85·21-s − 1.04·23-s − 2·27-s − 5/7·28-s + 3.29·31-s + 2.18·33-s − 7/6·36-s − 0.594·37-s − 0.651·43-s + 6/11·44-s + 2.80·47-s + 49-s − 5.64·51-s − 2.26·53-s + 4.42·57-s − 0.406·59-s − 1.18·61-s + 10/3·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01468754683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01468754683\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 10 T + 51 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 + 6 T - 85 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 48 T + 1514 T^{2} - 35808 T^{3} + 694947 T^{4} - 35808 p^{2} T^{5} + 1514 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 42 T + 1433 T^{2} + 35490 T^{3} + 795972 T^{4} + 35490 p^{2} T^{5} + 1433 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 24 T + 22 T^{2} - 12096 T^{3} - 277629 T^{4} - 12096 p^{2} T^{5} + 22 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 102 T + 6041 T^{2} - 8466 p T^{3} + 9396 p^{2} T^{4} - 8466 p^{3} T^{5} + 6041 p^{4} T^{6} - 102 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 22 T - 2087 T^{2} - 3674 T^{3} + 4073284 T^{4} - 3674 p^{2} T^{5} - 2087 p^{4} T^{6} + 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 3675 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T + 11654 T^{2} - 771672 T^{3} + 42125907 T^{4} - 771672 p^{2} T^{5} + 11654 p^{4} T^{6} - 132 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 120 T + 110 p T^{2} + 354240 T^{3} + 25104819 T^{4} + 354240 p^{2} T^{5} + 110 p^{5} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 6026 T^{2} + 140016 T^{3} + 22586547 T^{4} + 140016 p^{2} T^{5} + 6026 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 539280 p^{2} T^{5} + 9218 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 110 T + 3625 T^{2} - 55330 T^{3} - 2642396 T^{4} - 55330 p^{2} T^{5} + 3625 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 156 T + 178 p T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 66 T + 2873 T^{2} - 93786 T^{3} - 18641292 T^{4} - 93786 p^{2} T^{5} + 2873 p^{4} T^{6} - 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 10 T - 3695 T^{2} - 86870 T^{3} - 25172156 T^{4} - 86870 p^{2} T^{5} - 3695 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 36 T + 8353 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.79400476556162287912653337171, −6.71194662442022795916198891626, −6.41103152225436121972839590276, −5.99237458461935296650770489821, −5.95711397906995359105469548077, −5.85964530137194389945831537455, −5.52528098995266827957616892796, −5.24883035471263320362371983261, −5.03333917412405149341693951036, −4.83695083046739271996152300861, −4.79134472636754897611101344435, −4.52233213244292252338028260725, −4.08717860389135144045866247312, −3.91879207265715717443316867654, −3.90660336074076932526908622176, −3.22865684946373180334805760597, −3.09093433431019146863248351869, −2.56368539806353365034816642355, −2.46240205158143096245942205697, −1.95328285233156467160214126113, −1.73054114101247446634974450413, −1.11493823110572250908218057583, −1.09990514676108090002932623513, −0.70526034254819769807356385531, −0.02569986684135897953863664692,
0.02569986684135897953863664692, 0.70526034254819769807356385531, 1.09990514676108090002932623513, 1.11493823110572250908218057583, 1.73054114101247446634974450413, 1.95328285233156467160214126113, 2.46240205158143096245942205697, 2.56368539806353365034816642355, 3.09093433431019146863248351869, 3.22865684946373180334805760597, 3.90660336074076932526908622176, 3.91879207265715717443316867654, 4.08717860389135144045866247312, 4.52233213244292252338028260725, 4.79134472636754897611101344435, 4.83695083046739271996152300861, 5.03333917412405149341693951036, 5.24883035471263320362371983261, 5.52528098995266827957616892796, 5.85964530137194389945831537455, 5.95711397906995359105469548077, 5.99237458461935296650770489821, 6.41103152225436121972839590276, 6.71194662442022795916198891626, 6.79400476556162287912653337171
