| L(s) = 1 | + 36·3-s − 75·4-s + 810·9-s − 2.70e3·12-s + 2.57e3·16-s − 384·17-s − 2.42e3·23-s − 6.62e3·25-s + 1.45e4·27-s − 3.96e3·29-s − 6.07e4·36-s − 4.83e3·43-s + 9.26e4·48-s − 4.91e4·49-s − 1.38e4·51-s − 1.82e3·53-s − 1.00e5·61-s − 4.08e4·64-s + 2.88e4·68-s − 8.72e4·69-s − 2.38e5·75-s − 2.05e5·79-s + 2.29e5·81-s − 1.42e5·87-s + 1.81e5·92-s + 4.96e5·100-s − 1.09e5·101-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 2.34·4-s + 10/3·9-s − 5.41·12-s + 2.51·16-s − 0.322·17-s − 0.955·23-s − 2.11·25-s + 3.84·27-s − 0.874·29-s − 7.81·36-s − 0.398·43-s + 5.80·48-s − 2.92·49-s − 0.744·51-s − 0.0891·53-s − 3.47·61-s − 1.24·64-s + 0.755·68-s − 2.20·69-s − 4.89·75-s − 3.70·79-s + 35/9·81-s − 2.01·87-s + 2.23·92-s + 4.96·100-s − 1.06·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + 75 T^{2} + 763 p^{2} T^{4} + 75 p^{10} T^{6} + p^{20} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 6624 T^{2} + 1083838 p^{2} T^{4} + 6624 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 49132 T^{2} + 1140403638 T^{4} + 49132 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 617088 T^{2} + 146890430542 T^{4} + 617088 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 192 T + 2791006 T^{2} + 192 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 4292764 T^{2} + 9023685324870 T^{4} + 4292764 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 1212 T + 3450766 T^{2} + 1212 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 1980 T + 13967182 T^{2} + 1980 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 72558796 T^{2} + 2754768158742 p^{2} T^{4} + 72558796 p^{10} T^{6} + p^{20} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 191019364 T^{2} + 17466588618543606 T^{4} + 191019364 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 259347072 T^{2} + 41835734180463454 T^{4} + 259347072 p^{10} T^{6} + p^{20} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2416 T + 284123046 T^{2} + 2416 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 407420832 T^{2} + 124761975513464638 T^{4} + 407420832 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 912 T + 836540998 T^{2} + 912 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 151308144 T^{2} + 442844399562129262 T^{4} + 151308144 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 50496 T + 2324330710 T^{2} + 50496 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 45114940 T^{2} - 516263867957300538 T^{4} + 45114940 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 6784475376 T^{2} + 18015098428504544062 T^{4} + 6784475376 p^{10} T^{6} + p^{20} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 4119671084 T^{2} + 12172896815453134278 T^{4} - 4119671084 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 102672 T + 7610525470 T^{2} + 102672 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 12846419184 T^{2} + 72288762441539676238 T^{4} + 12846419184 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 9461922432 T^{2} + 70529599078905653662 T^{4} - 9461922432 p^{10} T^{6} + p^{20} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 23763592180 T^{2} + \)\(28\!\cdots\!94\)\( T^{4} + 23763592180 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.74801649045901333784392801056, −7.44811322610504661819340729846, −7.38163573698105284800302209147, −6.92100516634926891570087550570, −6.55702298762806375092085244157, −6.28894623424670590928748329723, −6.03981277417992113384897207670, −5.86583392115102769707981550020, −5.58032228664999910733436353820, −4.92541861615022497900352815993, −4.83535938370349450955982262206, −4.75433636393628351568335017217, −4.66495139239275351888990792239, −3.86308781691520581035982841596, −3.85120062670764233913457675068, −3.84819342393217094774425732744, −3.76807799652519818791088130831, −3.11567545038728981620952067849, −2.91753477510260959326940960539, −2.54928092323202099861618419884, −2.31417550677066230452095890941, −1.92380914613366125255712280273, −1.40089697489659642680782131982, −1.35362932424085345363593553894, −1.23113936034732143505098508850, 0, 0, 0, 0,
1.23113936034732143505098508850, 1.35362932424085345363593553894, 1.40089697489659642680782131982, 1.92380914613366125255712280273, 2.31417550677066230452095890941, 2.54928092323202099861618419884, 2.91753477510260959326940960539, 3.11567545038728981620952067849, 3.76807799652519818791088130831, 3.84819342393217094774425732744, 3.85120062670764233913457675068, 3.86308781691520581035982841596, 4.66495139239275351888990792239, 4.75433636393628351568335017217, 4.83535938370349450955982262206, 4.92541861615022497900352815993, 5.58032228664999910733436353820, 5.86583392115102769707981550020, 6.03981277417992113384897207670, 6.28894623424670590928748329723, 6.55702298762806375092085244157, 6.92100516634926891570087550570, 7.38163573698105284800302209147, 7.44811322610504661819340729846, 7.74801649045901333784392801056
