| L(s) = 1 | − 12·3-s + 90·9-s − 12·13-s + 96·17-s + 288·23-s + 240·25-s − 540·27-s − 384·29-s + 144·39-s + 1.28e3·43-s + 592·49-s − 1.15e3·51-s + 984·53-s + 288·61-s − 3.45e3·69-s − 2.88e3·75-s − 4.32e3·79-s + 2.83e3·81-s + 4.60e3·87-s + 1.19e3·103-s + 2.49e3·107-s − 1.08e3·117-s + 2.17e3·121-s + 127-s − 1.54e4·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 10/3·9-s − 0.256·13-s + 1.36·17-s + 2.61·23-s + 1.91·25-s − 3.84·27-s − 2.45·29-s + 0.591·39-s + 4.56·43-s + 1.72·49-s − 3.16·51-s + 2.55·53-s + 0.604·61-s − 6.02·69-s − 4.43·75-s − 6.15·79-s + 35/9·81-s + 5.67·87-s + 1.14·103-s + 2.25·107-s − 0.853·117-s + 1.63·121-s + 0.000698·127-s − 10.5·129-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.411269431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.411269431\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 13 | $D_{4}$ | \( 1 + 12 T - 74 p T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 48 p T^{2} + 44302 T^{4} - 48 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 592 T^{2} + 310782 T^{4} - 592 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 2172 T^{2} + 3811270 T^{4} - 2172 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 48 T - 1730 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 13552 T^{2} + 94862766 T^{4} - 13552 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 + 192 T + 45862 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 18592 T^{2} + 1722523710 T^{4} - 18592 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 92356 T^{2} + 6285341910 T^{4} - 92356 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 142320 T^{2} + 14548520830 T^{4} - 142320 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 644 T + 250566 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 385164 T^{2} + 58535996374 T^{4} - 385164 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 492 T + 164158 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 758940 T^{2} + 227837028550 T^{4} - 758940 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 144 T + 393094 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 611104 T^{2} + 187596510414 T^{4} - 611104 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 514188 T^{2} + 171350572726 T^{4} - 514188 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 659668 T^{2} + 389934034566 T^{4} - 659668 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 2160 T + 2130910 T^{2} + 2160 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 823068 T^{2} + 604381635622 T^{4} - 823068 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 2258064 T^{2} + 2268670823038 T^{4} - 2258064 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 3513652 T^{2} + 4748412207846 T^{4} - 3513652 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−7.21605549954595346516232100556, −6.96081526191463969324178481706, −6.95186559264731397693035766090, −6.52458451392073934965329324215, −5.99173331311633127457075692325, −5.94648836305101518383354422432, −5.62629711219449664054645075919, −5.60326446300382633811894292562, −5.41845105668020644916392305368, −5.28453503870912372585151838280, −4.79418730344071520372774672993, −4.45378985298353026462057350218, −4.29750042132012070372497520901, −4.22556313521434607967346895397, −3.85403956890721691057494654794, −3.32190541622687834978804157490, −3.13900691953855585867294867529, −2.80191696755679826239559333043, −2.42746807941748610970659952748, −2.12142288519310353278334880561, −1.47473747671249465120200783535, −1.21555590437516458524483464041, −0.908052569580077579108640718754, −0.61200891715028187028052419052, −0.45458263623674012964068728426,
0.45458263623674012964068728426, 0.61200891715028187028052419052, 0.908052569580077579108640718754, 1.21555590437516458524483464041, 1.47473747671249465120200783535, 2.12142288519310353278334880561, 2.42746807941748610970659952748, 2.80191696755679826239559333043, 3.13900691953855585867294867529, 3.32190541622687834978804157490, 3.85403956890721691057494654794, 4.22556313521434607967346895397, 4.29750042132012070372497520901, 4.45378985298353026462057350218, 4.79418730344071520372774672993, 5.28453503870912372585151838280, 5.41845105668020644916392305368, 5.60326446300382633811894292562, 5.62629711219449664054645075919, 5.94648836305101518383354422432, 5.99173331311633127457075692325, 6.52458451392073934965329324215, 6.95186559264731397693035766090, 6.96081526191463969324178481706, 7.21605549954595346516232100556
