| L(s) = 1 | + 4.37e3·3-s + 6.96e4·5-s + 2.49e6·7-s + 1.43e7·9-s − 1.49e7·11-s + 1.17e7·13-s + 3.04e8·15-s + 4.25e9·17-s + 9.24e9·19-s + 1.08e10·21-s + 2.42e10·23-s + 9.10e9·25-s + 4.18e10·27-s + 1.47e11·29-s − 7.66e10·31-s − 6.55e10·33-s + 1.73e11·35-s − 9.71e11·37-s + 5.14e10·39-s − 2.01e12·41-s + 6.98e11·43-s + 9.99e11·45-s − 7.83e11·47-s + 5.46e11·49-s + 1.86e13·51-s − 5.58e12·53-s − 1.04e12·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.398·5-s + 1.14·7-s + 9-s − 0.231·11-s + 0.0519·13-s + 0.460·15-s + 2.51·17-s + 2.37·19-s + 1.32·21-s + 1.48·23-s + 0.298·25-s + 0.769·27-s + 1.58·29-s − 0.500·31-s − 0.267·33-s + 0.455·35-s − 1.68·37-s + 0.0600·39-s − 1.61·41-s + 0.392·43-s + 0.398·45-s − 0.225·47-s + 0.115·49-s + 2.90·51-s − 0.653·53-s − 0.0923·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(6.775473040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.775473040\) |
| \(L(\frac{17}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{7} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 13932 p T - 169916786 p^{2} T^{2} - 13932 p^{16} T^{3} + p^{30} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2491504 T + 808689782370 p T^{2} - 2491504 p^{15} T^{3} + p^{30} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 123768 p^{2} T + 24557375007238 p^{2} T^{2} + 123768 p^{17} T^{3} + p^{30} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 11757580 T - 954287309447322 p T^{2} - 11757580 p^{15} T^{3} + p^{30} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4256077284 T + 10157958945751056550 T^{2} - 4256077284 p^{15} T^{3} + p^{30} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9241689400 T + 49788335786742451398 T^{2} - 9241689400 p^{15} T^{3} + p^{30} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 24254822736 T + \)\(68\!\cdots\!38\)\( T^{2} - 24254822736 p^{15} T^{3} + p^{30} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5078974428 p T + \)\(22\!\cdots\!34\)\( T^{2} - 5078974428 p^{16} T^{3} + p^{30} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 76677530432 T + \)\(34\!\cdots\!58\)\( T^{2} + 76677530432 p^{15} T^{3} + p^{30} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 971233156388 T + \)\(89\!\cdots\!22\)\( T^{2} + 971233156388 p^{15} T^{3} + p^{30} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2010703420620 T + \)\(36\!\cdots\!02\)\( T^{2} + 2010703420620 p^{15} T^{3} + p^{30} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 698966302216 T + \)\(13\!\cdots\!78\)\( T^{2} - 698966302216 p^{15} T^{3} + p^{30} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 783844820064 T + \)\(23\!\cdots\!10\)\( T^{2} + 783844820064 p^{15} T^{3} + p^{30} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5585874545988 T + \)\(67\!\cdots\!50\)\( T^{2} + 5585874545988 p^{15} T^{3} + p^{30} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15836162237208 T + \)\(58\!\cdots\!14\)\( T^{2} + 15836162237208 p^{15} T^{3} + p^{30} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1816274061484 T + \)\(11\!\cdots\!66\)\( T^{2} - 1816274061484 p^{15} T^{3} + p^{30} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2258067410984 T + \)\(20\!\cdots\!50\)\( T^{2} + 2258067410984 p^{15} T^{3} + p^{30} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 196995319710480 T + \)\(21\!\cdots\!02\)\( T^{2} + 196995319710480 p^{15} T^{3} + p^{30} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8025857227532 T - \)\(54\!\cdots\!30\)\( T^{2} + 8025857227532 p^{15} T^{3} + p^{30} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 356386066144864 T + \)\(83\!\cdots\!22\)\( T^{2} - 356386066144864 p^{15} T^{3} + p^{30} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 361504357486776 T + \)\(12\!\cdots\!58\)\( T^{2} - 361504357486776 p^{15} T^{3} + p^{30} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 184456308480300 T + \)\(18\!\cdots\!98\)\( T^{2} + 184456308480300 p^{15} T^{3} + p^{30} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1815571461080252 T + \)\(20\!\cdots\!62\)\( T^{2} + 1815571461080252 p^{15} T^{3} + p^{30} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.61656234131841306397386931253, −16.05149788956371203891494909018, −15.07753628230668599189688278737, −14.60279810102225804673490194956, −13.79027085925140220645961112575, −13.75308788890089527576574700247, −12.34810128115876150373240841919, −11.92197693906859573211775253305, −10.73145951406753837623981644916, −9.965821580257841470669893056781, −9.291217643086866711479017907019, −8.331241114772078929825742419517, −7.73680689795019823313762682362, −7.02969759536102160285976887270, −5.37771471437319157369458385082, −4.96864991170780892777131520898, −3.33925771476485868183658896504, −2.98290998870518709035005154944, −1.44710926774454760527022656313, −1.17335982733503466991441904344,
1.17335982733503466991441904344, 1.44710926774454760527022656313, 2.98290998870518709035005154944, 3.33925771476485868183658896504, 4.96864991170780892777131520898, 5.37771471437319157369458385082, 7.02969759536102160285976887270, 7.73680689795019823313762682362, 8.331241114772078929825742419517, 9.291217643086866711479017907019, 9.965821580257841470669893056781, 10.73145951406753837623981644916, 11.92197693906859573211775253305, 12.34810128115876150373240841919, 13.75308788890089527576574700247, 13.79027085925140220645961112575, 14.60279810102225804673490194956, 15.07753628230668599189688278737, 16.05149788956371203891494909018, 16.61656234131841306397386931253
