| 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
| \(L(2)\) |
\(\approx\) |
\(0.09646782838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09646782838\) |
| \(L(\frac{5}{2})\) |
|
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
−8.873181406561208984578249387727, −8.460314949510064302249630975170, −8.144534801621012939231831879453, −8.102330198186100131230183158952, −7.959670871159863295939927607386, −7.13373958230152592505366690610, −6.95788491909068058194357041145, −6.83234721273674783448391046187, −6.68602040452028694379964110969, −6.22630619640177156132516453931, −6.18742829622755148770092677614, −5.81336225753821492746555505449, −5.51985043016919928253989513626, −5.40891434115284325877930497151, −4.53755512544932741186045453364, −4.26560926106881548762164216511, −4.07775824836160001993222480800, −3.99827115024216889201159326961, −3.35580831474923524967147856710, −2.69148743099047099217927677755, −2.57945313602027057362813716387, −1.61847531322624449971105759598, −1.59441261236114764025577277313, −1.52746612954203805817548679790, −0.05705526035593927403444570681,
0.05705526035593927403444570681, 1.52746612954203805817548679790, 1.59441261236114764025577277313, 1.61847531322624449971105759598, 2.57945313602027057362813716387, 2.69148743099047099217927677755, 3.35580831474923524967147856710, 3.99827115024216889201159326961, 4.07775824836160001993222480800, 4.26560926106881548762164216511, 4.53755512544932741186045453364, 5.40891434115284325877930497151, 5.51985043016919928253989513626, 5.81336225753821492746555505449, 6.18742829622755148770092677614, 6.22630619640177156132516453931, 6.68602040452028694379964110969, 6.83234721273674783448391046187, 6.95788491909068058194357041145, 7.13373958230152592505366690610, 7.959670871159863295939927607386, 8.102330198186100131230183158952, 8.144534801621012939231831879453, 8.460314949510064302249630975170, 8.873181406561208984578249387727
