| L(s) = 1 | + 4·3-s + 4·5-s + 10·7-s + 11·9-s − 2·11-s − 8·13-s + 16·15-s + 8·17-s + 2·19-s + 40·21-s − 14·23-s + 5·25-s + 20·27-s + 12·31-s − 8·33-s + 40·35-s − 12·37-s − 32·39-s + 6·43-s + 44·45-s − 6·47-s + 61·49-s + 32·51-s + 10·53-s − 8·55-s + 8·57-s + 6·59-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s + 3.77·7-s + 11/3·9-s − 0.603·11-s − 2.21·13-s + 4.13·15-s + 1.94·17-s + 0.458·19-s + 8.72·21-s − 2.91·23-s + 25-s + 3.84·27-s + 2.15·31-s − 1.39·33-s + 6.76·35-s − 1.97·37-s − 5.12·39-s + 0.914·43-s + 6.55·45-s − 0.875·47-s + 61/7·49-s + 4.48·51-s + 1.37·53-s − 1.07·55-s + 1.05·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.58883708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.58883708\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 4 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 20 T^{2} + 52 T^{3} - 545 T^{4} + 52 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 4 p T^{5} - 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 14 T + 53 T^{2} - 226 T^{3} - 2552 T^{4} - 226 p T^{5} + 53 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 696 p T^{5} + 106 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 60 T^{3} - 889 T^{4} - 60 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T + 90 T^{2} + 672 T^{3} + 5159 T^{4} + 672 p T^{5} + 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 108 p T^{5} - 64 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 1308 p T^{5} + 157 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 137 T^{2} + 1224 T^{3} + 11492 T^{4} + 1224 p T^{5} + 137 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 144 T^{2} + 600 T^{3} + 10991 T^{4} + 600 p T^{5} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 13203 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 140 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.004213607318069454388351241156, −7.48071556414062015574682885705, −7.47540805662847755451117666927, −7.27375612590516772429018853883, −7.26738058745607335418650563103, −6.62309151778698760282845053007, −6.21199058081073731396058415306, −6.08156285394035009559105904264, −5.73541475843463463203343388238, −5.23405700479896909505779315334, −5.14043367461765199068711254715, −5.12271736934294828556066785657, −5.06807857966314492239820865664, −4.31324408447124397099924329877, −4.14982750212631766048856426621, −4.08737456179851799988518545782, −3.82859304990681205797880679444, −3.09642107649895656094075037443, −2.66028460896628459813390347535, −2.63841316684878581418881363766, −2.38552181202392545504218212042, −1.74302988403643170676739164818, −1.73320803368637708840705051486, −1.70258881407282085513826207091, −1.10255016746565741358781149159,
1.10255016746565741358781149159, 1.70258881407282085513826207091, 1.73320803368637708840705051486, 1.74302988403643170676739164818, 2.38552181202392545504218212042, 2.63841316684878581418881363766, 2.66028460896628459813390347535, 3.09642107649895656094075037443, 3.82859304990681205797880679444, 4.08737456179851799988518545782, 4.14982750212631766048856426621, 4.31324408447124397099924329877, 5.06807857966314492239820865664, 5.12271736934294828556066785657, 5.14043367461765199068711254715, 5.23405700479896909505779315334, 5.73541475843463463203343388238, 6.08156285394035009559105904264, 6.21199058081073731396058415306, 6.62309151778698760282845053007, 7.26738058745607335418650563103, 7.27375612590516772429018853883, 7.47540805662847755451117666927, 7.48071556414062015574682885705, 8.004213607318069454388351241156
