| L(s) = 1 | + 2·2-s + 4-s + 2·5-s − 2·7-s − 2·8-s + 4·10-s − 11-s − 2·13-s − 4·14-s − 4·16-s − 8·17-s + 19-s + 2·20-s − 2·22-s + 6·23-s − 9·25-s − 4·26-s − 2·28-s − 4·29-s − 2·32-s − 16·34-s − 4·35-s + 9·37-s + 2·38-s − 4·40-s − 6·41-s + 16·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s − 0.707·8-s + 1.26·10-s − 0.301·11-s − 0.554·13-s − 1.06·14-s − 16-s − 1.94·17-s + 0.229·19-s + 0.447·20-s − 0.426·22-s + 1.25·23-s − 9/5·25-s − 0.784·26-s − 0.377·28-s − 0.742·29-s − 0.353·32-s − 2.74·34-s − 0.676·35-s + 1.47·37-s + 0.324·38-s − 0.632·40-s − 0.937·41-s + 2.43·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.269171957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.269171957\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + T - 17 T^{2} - 4 T^{3} + 192 T^{4} - 4 p T^{5} - 17 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 31 T^{2} - 8 T^{3} - 288 T^{4} - 8 p T^{5} + 31 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - T - 33 T^{2} + 4 T^{3} + 776 T^{4} + 4 p T^{5} - 33 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 2 T^{2} + 48 T^{3} + 87 T^{4} + 48 p T^{5} - 2 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T - p T^{2} - 52 T^{3} + 720 T^{4} - 52 p T^{5} - p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 9 T - 9 T^{2} - 144 T^{3} + 3734 T^{4} - 144 p T^{5} - 9 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T + 13 T^{2} - 354 T^{3} - 2628 T^{4} - 354 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 2 T - 98 T^{2} + 32 T^{3} + 6687 T^{4} + 32 p T^{5} - 98 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 27 T^{2} + 354 T^{3} - 1948 T^{4} + 354 p T^{5} - 27 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 42 T^{2} + 80 T^{3} + 8975 T^{4} + 80 p T^{5} - 42 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 2 T + 14 T^{2} + 304 T^{3} - 5225 T^{4} + 304 p T^{5} + 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 2 T + 150 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 11 T - 83 T^{2} - 286 T^{3} + 22926 T^{4} - 286 p T^{5} - 83 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 10 T - 102 T^{2} - 80 T^{3} + 21695 T^{4} - 80 p T^{5} - 102 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.58093450482566888927716728019, −6.19742267115986049732217095851, −6.11312163058295624448051015148, −5.95949465329718796961337749226, −5.86765774386238263638022684472, −5.83291960401309929747625185446, −5.30127145324480245566499110128, −5.13735850246091833563259639372, −4.86465280695480611237745380927, −4.81514980347337709897850752316, −4.44791380782815948558392714458, −4.19742672576276954068705998488, −4.17202646854087974721956256394, −3.82207054287793090124277997255, −3.57676029877861558621801025528, −3.42195206903390873983348987343, −2.77191939724509779611945423332, −2.76579457277474060468395235495, −2.69501353165692838753041901982, −2.21936010673044061907192019995, −2.18853298098297566924413809656, −1.58731383008501234063753043307, −1.37520253623253776125545063749, −0.74637685495659291268656313059, −0.17747955018608969153414201622,
0.17747955018608969153414201622, 0.74637685495659291268656313059, 1.37520253623253776125545063749, 1.58731383008501234063753043307, 2.18853298098297566924413809656, 2.21936010673044061907192019995, 2.69501353165692838753041901982, 2.76579457277474060468395235495, 2.77191939724509779611945423332, 3.42195206903390873983348987343, 3.57676029877861558621801025528, 3.82207054287793090124277997255, 4.17202646854087974721956256394, 4.19742672576276954068705998488, 4.44791380782815948558392714458, 4.81514980347337709897850752316, 4.86465280695480611237745380927, 5.13735850246091833563259639372, 5.30127145324480245566499110128, 5.83291960401309929747625185446, 5.86765774386238263638022684472, 5.95949465329718796961337749226, 6.11312163058295624448051015148, 6.19742267115986049732217095851, 6.58093450482566888927716728019
