| L(s) = 1 | − 2-s − 5·3-s + 12·4-s − 6·5-s + 5·6-s + 9·7-s − 19·8-s + 22·9-s + 6·10-s − 80·11-s − 60·12-s − 9·14-s + 30·15-s + 75·16-s − 19·17-s − 22·18-s + 84·19-s − 72·20-s − 45·21-s + 80·22-s − 196·23-s + 95·24-s − 469·25-s + 65·27-s + 108·28-s + 44·29-s − 30·30-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.962·3-s + 3/2·4-s − 0.536·5-s + 0.340·6-s + 0.485·7-s − 0.839·8-s + 0.814·9-s + 0.189·10-s − 2.19·11-s − 1.44·12-s − 0.171·14-s + 0.516·15-s + 1.17·16-s − 0.271·17-s − 0.288·18-s + 1.01·19-s − 0.804·20-s − 0.467·21-s + 0.775·22-s − 1.77·23-s + 0.807·24-s − 3.75·25-s + 0.463·27-s + 0.728·28-s + 0.281·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.543485254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.543485254\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T - 11 T^{2} - p^{2} T^{3} + 9 p^{3} T^{4} - p^{5} T^{5} - 11 p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 5 T + p T^{2} - 160 T^{3} - 920 T^{4} - 160 p^{3} T^{5} + p^{7} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 3 T + 248 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 9 T - 111 T^{2} + 4446 T^{3} - 108568 T^{4} + 4446 p^{3} T^{5} - 111 p^{6} T^{6} - 9 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 80 T + 250 p T^{2} + 79040 T^{3} + 3032539 T^{4} + 79040 p^{3} T^{5} + 250 p^{7} T^{6} + 80 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 19 T - 8327 T^{2} - 21622 T^{3} + 49570182 T^{4} - 21622 p^{3} T^{5} - 8327 p^{6} T^{6} + 19 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 84 T - 4074 T^{2} + 217392 T^{3} + 28433915 T^{4} + 217392 p^{3} T^{5} - 4074 p^{6} T^{6} - 84 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 196 T + 5090 T^{2} + 1762432 T^{3} + 495178915 T^{4} + 1762432 p^{3} T^{5} + 5090 p^{6} T^{6} + 196 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 44 T - 8158 T^{2} + 1702096 T^{3} - 540151589 T^{4} + 1702096 p^{3} T^{5} - 8158 p^{6} T^{6} - 44 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 86 T + 56518 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 209 T - 68439 T^{2} + 2260126 T^{3} + 7792594298 T^{4} + 2260126 p^{3} T^{5} - 68439 p^{6} T^{6} + 209 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 230 T - 96110 T^{2} - 2568640 T^{3} + 13807954959 T^{4} - 2568640 p^{3} T^{5} - 96110 p^{6} T^{6} - 230 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 287 T - 10329 T^{2} - 19032692 T^{3} - 4277355928 T^{4} - 19032692 p^{3} T^{5} - 10329 p^{6} T^{6} + 287 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 435 T + 192728 T^{2} - 435 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 118 T + 297410 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 368 T - 243842 T^{2} + 11589056 T^{3} + 73848919419 T^{4} + 11589056 p^{3} T^{5} - 243842 p^{6} T^{6} - 368 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 1058 T + 538986 T^{2} - 133748128 T^{3} + 31243888439 T^{4} - 133748128 p^{3} T^{5} + 538986 p^{6} T^{6} - 1058 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 68 T - 369306 T^{2} - 15476528 T^{3} + 47974534619 T^{4} - 15476528 p^{3} T^{5} - 369306 p^{6} T^{6} + 68 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 131 T - 476167 T^{2} + 29146714 T^{3} + 109130120992 T^{4} + 29146714 p^{3} T^{5} - 476167 p^{6} T^{6} - 131 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 456 T + 542718 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 1958 T + 1961238 T^{2} - 1958 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 720 T - 381326 T^{2} + 367352640 T^{3} - 52930345485 T^{4} + 367352640 p^{3} T^{5} - 381326 p^{6} T^{6} - 720 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 928 T - 82686 T^{2} + 818009728 T^{3} - 728060812861 T^{4} + 818009728 p^{3} T^{5} - 82686 p^{6} T^{6} - 928 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−8.872548326051228678917894994730, −8.208238680157439429258784355755, −8.195272314577383850875855156911, −8.192825342194267510667059050206, −7.67291323812057560449669328499, −7.42982581603841182947961645868, −7.41620704505801435621887793312, −6.99527308058620390429088172851, −6.63436948398540763888466295986, −6.37051195911218099393106461508, −5.92890293004411955676427205076, −5.69854408767866630294576163817, −5.44835332201661153865513189541, −5.39699266518487194503548511801, −4.86133908838088317469736738659, −4.42265158302877262761031837152, −3.97300152064603835538714524139, −3.65808196524719961611417011260, −3.44947885884092282924320216215, −2.54321628636650256424530971474, −2.45259118933249809942261512364, −1.95652601494146955328175586963, −1.79768339759199446983018655371, −0.63462084242973912059913346163, −0.44194807365249133003897886939,
0.44194807365249133003897886939, 0.63462084242973912059913346163, 1.79768339759199446983018655371, 1.95652601494146955328175586963, 2.45259118933249809942261512364, 2.54321628636650256424530971474, 3.44947885884092282924320216215, 3.65808196524719961611417011260, 3.97300152064603835538714524139, 4.42265158302877262761031837152, 4.86133908838088317469736738659, 5.39699266518487194503548511801, 5.44835332201661153865513189541, 5.69854408767866630294576163817, 5.92890293004411955676427205076, 6.37051195911218099393106461508, 6.63436948398540763888466295986, 6.99527308058620390429088172851, 7.41620704505801435621887793312, 7.42982581603841182947961645868, 7.67291323812057560449669328499, 8.192825342194267510667059050206, 8.195272314577383850875855156911, 8.208238680157439429258784355755, 8.872548326051228678917894994730
