| L(s) = 1 | − 8·2-s + 40·4-s − 160·8-s + 560·16-s + 234·17-s − 161·25-s + 222·29-s − 242·31-s − 1.79e3·32-s − 1.87e3·34-s − 368·37-s − 42·41-s − 1.03e3·49-s + 1.28e3·50-s − 1.77e3·58-s + 1.93e3·62-s + 5.37e3·64-s − 1.03e3·67-s + 9.36e3·68-s + 2.94e3·74-s + 336·82-s + 1.05e3·83-s − 3.08e3·97-s + 8.26e3·98-s − 6.44e3·100-s − 876·101-s − 3.73e3·103-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 7.07·8-s + 35/4·16-s + 3.33·17-s − 1.28·25-s + 1.42·29-s − 1.40·31-s − 9.89·32-s − 9.44·34-s − 1.63·37-s − 0.159·41-s − 3.01·49-s + 3.64·50-s − 4.02·58-s + 3.96·62-s + 21/2·64-s − 1.88·67-s + 16.6·68-s + 4.62·74-s + 0.452·82-s + 1.39·83-s − 3.23·97-s + 8.51·98-s − 6.43·100-s − 0.863·101-s − 3.56·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6336772308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6336772308\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 161 T^{2} + 26424 T^{4} + 161 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 1033 T^{2} + 490764 T^{4} + 1033 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 7825 T^{2} + 24761832 T^{4} + 7825 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 117 T + 9178 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 3769 T^{2} - 33745020 T^{4} + 3769 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 26972 T^{2} + 431633574 T^{4} + 26972 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 111 T + 29698 T^{2} - 111 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 121 T + 41082 T^{2} + 121 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 184 T + 93489 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 21 T + 126646 T^{2} + 21 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 174928 T^{2} + 15856365390 T^{4} + 174928 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 366476 T^{2} + 54900148902 T^{4} + 366476 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 405833 T^{2} + 77238676824 T^{4} + 405833 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 772316 T^{2} + 233264998422 T^{4} + 772316 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 58489 T^{2} + 101760497112 T^{4} + 58489 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 517 T + 49218 T^{2} + 517 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 945404 T^{2} + 424845873702 T^{4} + 945404 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 1281265 T^{2} + 707692426032 T^{4} + 1281265 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 963385 T^{2} + 710655778092 T^{4} + 963385 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 528 T + 1184326 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 160759 T^{2} - 350424876528 T^{4} - 160759 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 1544 T + 1898529 T^{2} + 1544 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.34477908060371137789539802857, −5.82216653037713417066812072997, −5.69713624596410364288554426581, −5.54739678109715330292824259371, −5.53980503250301980445881838503, −5.13444904782014945304926780666, −4.91809513360947420906282270959, −4.57797865254072939286274448004, −4.50856078126550592909986960378, −3.72131540184198831961531751977, −3.71158364923209030194037701622, −3.64493232434709260690367787847, −3.49268229891361797105289169347, −2.90732068951028681260757085261, −2.86642606950959255013195629451, −2.68957756516415896751906380832, −2.49820724606955243890945826032, −1.74249802933826129550567594618, −1.66355119372383539124669185735, −1.63364169415534674370649678062, −1.46080883314845094916198554674, −1.04584586671652264580759405814, −0.75414813008863053067621536136, −0.34890956025031360875340771987, −0.21853064220122109807148569084,
0.21853064220122109807148569084, 0.34890956025031360875340771987, 0.75414813008863053067621536136, 1.04584586671652264580759405814, 1.46080883314845094916198554674, 1.63364169415534674370649678062, 1.66355119372383539124669185735, 1.74249802933826129550567594618, 2.49820724606955243890945826032, 2.68957756516415896751906380832, 2.86642606950959255013195629451, 2.90732068951028681260757085261, 3.49268229891361797105289169347, 3.64493232434709260690367787847, 3.71158364923209030194037701622, 3.72131540184198831961531751977, 4.50856078126550592909986960378, 4.57797865254072939286274448004, 4.91809513360947420906282270959, 5.13444904782014945304926780666, 5.53980503250301980445881838503, 5.54739678109715330292824259371, 5.69713624596410364288554426581, 5.82216653037713417066812072997, 6.34477908060371137789539802857
