| L(s) = 1 | − 128·4-s + 4.35e3·9-s + 9.03e3·11-s + 1.22e4·16-s − 8.74e4·19-s − 1.85e4·29-s + 4.14e5·31-s − 5.57e5·36-s + 5.29e5·41-s − 1.15e6·44-s − 2.35e5·49-s − 3.81e6·59-s − 2.38e5·61-s − 1.04e6·64-s − 1.22e6·71-s + 1.11e7·76-s + 2.52e7·79-s + 4.71e6·81-s + 2.23e7·89-s + 3.93e7·99-s + 2.67e7·101-s + 3.12e7·109-s + 2.37e6·116-s + 4.59e7·121-s − 5.29e7·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s + 1.99·9-s + 2.04·11-s + 3/4·16-s − 2.92·19-s − 0.141·29-s + 2.49·31-s − 1.99·36-s + 1.20·41-s − 2.04·44-s − 2/7·49-s − 2.41·59-s − 0.134·61-s − 1/2·64-s − 0.406·71-s + 2.92·76-s + 5.75·79-s + 0.985·81-s + 3.36·89-s + 4.07·99-s + 2.57·101-s + 2.31·109-s + 0.141·116-s + 2.35·121-s − 2.49·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(12.54359670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.54359670\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4355 T^{2} + 1583632 p^{2} T^{4} - 4355 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4515 T + 7622698 T^{2} - 4515 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 199575983 T^{2} + 17376370741926888 T^{4} - 199575983 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 912351335 T^{2} + 527496167207941008 T^{4} - 912351335 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 43702 T + 2264500038 T^{2} + 43702 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2377783064 T^{2} + 15998568625312665198 T^{4} - 2377783064 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 9261 T + 29046530668 T^{2} + 9261 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 207028 T + 63353144094 T^{2} - 207028 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 90540717740 T^{2} + \)\(17\!\cdots\!78\)\( T^{4} - 90540717740 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 264798 T + 343312202314 T^{2} - 264798 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 949377556568 T^{2} + \)\(37\!\cdots\!38\)\( T^{4} - 949377556568 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1896066244187 T^{2} + \)\(14\!\cdots\!88\)\( T^{4} - 1896066244187 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4458983263808 T^{2} + \)\(77\!\cdots\!58\)\( T^{4} - 4458983263808 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 1905408 T + 5884867139878 T^{2} + 1905408 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 119318 T + 220389872202 T^{2} + 119318 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6680592800780 T^{2} + \)\(78\!\cdots\!58\)\( T^{4} - 6680592800780 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 612672 T - 10976915566226 T^{2} + 612672 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 34166737841564 T^{2} + \)\(51\!\cdots\!98\)\( T^{4} - 34166737841564 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 12620759 T + 76528557994518 T^{2} - 12620759 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 101228266399580 T^{2} + \)\(40\!\cdots\!58\)\( T^{4} - 101228266399580 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 11178330 T + 114686589507658 T^{2} - 11178330 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 223835956712591 T^{2} + \)\(23\!\cdots\!28\)\( T^{4} - 223835956712591 p^{14} T^{6} + p^{28} T^{8} \) |
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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
−7.24788958870918122534107285312, −6.48850210883995776802624341478, −6.44601968186503434112930024299, −6.44191220653227920277744851604, −6.23511386342487735867427030284, −6.19720654810147334139903902891, −5.51079409022242760297772670868, −5.13901202851118832812273705421, −4.67191753970485882366629250370, −4.65005456478097585417163664329, −4.62075966730449006153952891588, −4.21374631131315588638288775518, −4.01341486589629185275671842654, −3.71880819934961154682809656266, −3.55784632542891110961796891959, −3.19387196918847141705313523181, −2.67197266616893315015538703294, −2.39496923066067519558058989281, −1.82877078736798975723853622916, −1.72552978512923968753927950926, −1.72510030869690200513554946762, −1.01831747274337336092111507887, −0.75331753309352303785891048237, −0.56758273295647911442693825161, −0.46487262549327110866363308595,
0.46487262549327110866363308595, 0.56758273295647911442693825161, 0.75331753309352303785891048237, 1.01831747274337336092111507887, 1.72510030869690200513554946762, 1.72552978512923968753927950926, 1.82877078736798975723853622916, 2.39496923066067519558058989281, 2.67197266616893315015538703294, 3.19387196918847141705313523181, 3.55784632542891110961796891959, 3.71880819934961154682809656266, 4.01341486589629185275671842654, 4.21374631131315588638288775518, 4.62075966730449006153952891588, 4.65005456478097585417163664329, 4.67191753970485882366629250370, 5.13901202851118832812273705421, 5.51079409022242760297772670868, 6.19720654810147334139903902891, 6.23511386342487735867427030284, 6.44191220653227920277744851604, 6.44601968186503434112930024299, 6.48850210883995776802624341478, 7.24788958870918122534107285312
