| L(s) = 1 | + 2·4-s − 125·16-s + 144·17-s + 276·23-s − 310·25-s − 12·29-s + 940·43-s − 340·49-s + 2.26e3·53-s + 320·61-s − 380·64-s + 288·68-s + 8·79-s + 552·92-s − 620·100-s − 636·101-s − 116·103-s + 3.70e3·107-s + 1.70e3·113-s − 24·116-s − 2.64e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/4·4-s − 1.95·16-s + 2.05·17-s + 2.50·23-s − 2.47·25-s − 0.0768·29-s + 3.33·43-s − 0.991·49-s + 5.87·53-s + 0.671·61-s − 0.742·64-s + 0.513·68-s + 0.0113·79-s + 0.625·92-s − 0.619·100-s − 0.626·101-s − 0.110·103-s + 3.35·107-s + 1.41·113-s − 0.0192·116-s − 1.98·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.556755449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.556755449\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{6} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 62 p T^{2} + 47931 T^{4} + 62 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 340 T^{2} + 858 p^{2} T^{4} + 340 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2644 T^{2} + 5200842 T^{4} + 2644 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 72 T + 11071 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 16468 T^{2} + 139269162 T^{4} + 16468 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 p T + 26596 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 24103 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 22684 T^{2} - 285715578 T^{4} + 22684 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 93418 T^{2} + 4535499099 T^{4} + 93418 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 270250 T^{2} + 27758790507 T^{4} + 270250 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 470 T + 202764 T^{2} - 470 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 129172 T^{2} + 10043128410 T^{4} + 129172 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 1134 T + 602719 T^{2} - 1134 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 41740 T^{2} + 83104566582 T^{4} + 41740 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 160 T + 74343 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 1146724 T^{2} + 509040985386 T^{4} + 1146724 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 1134052 T^{2} + 574891463802 T^{4} + 1134052 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 626159 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 176406 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 124420 T^{2} + 286548422442 T^{4} + 124420 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 818116 T^{2} + 1140216422502 T^{4} + 818116 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1862660 T^{2} + 2429934389382 T^{4} - 1862660 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.42639117531139519019351153973, −6.01937795329651416388540407111, −5.91109963343907352625484927131, −5.80492455606930698474407119857, −5.56607778051098132518112166994, −5.13320023606361229951509780997, −5.11461393272137153443034943744, −4.95908746764577473682108019122, −4.72550328205543154914964092093, −4.06942430857133248660673100233, −4.04672663918203774942455230607, −3.95645860451255534333727268761, −3.88254955175204368082873557903, −3.41901348728940541595164949784, −3.09597516843740866887606803823, −2.76367935986095977221814498973, −2.59231736481393849662359922981, −2.35804987782895740355226355379, −2.27338577481491736693551178231, −1.72010638527191251343467174974, −1.56371757947904962003783036961, −0.991430142294612427054925713228, −0.934167160143925575694819025422, −0.47348700764199541273479608949, −0.43696791310155239604860135844,
0.43696791310155239604860135844, 0.47348700764199541273479608949, 0.934167160143925575694819025422, 0.991430142294612427054925713228, 1.56371757947904962003783036961, 1.72010638527191251343467174974, 2.27338577481491736693551178231, 2.35804987782895740355226355379, 2.59231736481393849662359922981, 2.76367935986095977221814498973, 3.09597516843740866887606803823, 3.41901348728940541595164949784, 3.88254955175204368082873557903, 3.95645860451255534333727268761, 4.04672663918203774942455230607, 4.06942430857133248660673100233, 4.72550328205543154914964092093, 4.95908746764577473682108019122, 5.11461393272137153443034943744, 5.13320023606361229951509780997, 5.56607778051098132518112166994, 5.80492455606930698474407119857, 5.91109963343907352625484927131, 6.01937795329651416388540407111, 6.42639117531139519019351153973
