| L(s) = 1 | − 88·25-s + 196·49-s − 568·73-s − 200·97-s − 680·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 100·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 3.51·25-s + 4·49-s − 7.78·73-s − 2.06·97-s − 5.61·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.591·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.006141648402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006141648402\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 22 T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 170 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 + 25 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 362 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 647 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 842 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 + 910 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1822 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2375 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 1970 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 + 982 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 - 5330 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 + 6314 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 7079 T^{2} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 + 3742 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 71 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 + 10319 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 4654 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 - 10294 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 25 T + p^{2} T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.76660575400644437269900305099, −3.59073577326330632765067038176, −3.42933872523375038331995076719, −3.42656809080920182652120076723, −3.41439945873293673447109025019, −2.91432149632903365904808842496, −2.84961421672070501153407091319, −2.82756636959243666788269527181, −2.69928626153238513022453264921, −2.53001806298188319935418489845, −2.47982372484162658418930824924, −2.34646427030698464644893233113, −2.12010177178148880065823325999, −2.08100580542859812053489934032, −1.84714489078494559952366711481, −1.58411404395506051792494799540, −1.49620344808295110670517619386, −1.41692900422705889589498668637, −1.25210031962785412223389780869, −1.13409147605806736278049902632, −1.02290904791605000695517208850, −0.57950369475835166081938675894, −0.44222130663436500459540019270, −0.17894393778318282928850578274, −0.01065779687532857165666677581,
0.01065779687532857165666677581, 0.17894393778318282928850578274, 0.44222130663436500459540019270, 0.57950369475835166081938675894, 1.02290904791605000695517208850, 1.13409147605806736278049902632, 1.25210031962785412223389780869, 1.41692900422705889589498668637, 1.49620344808295110670517619386, 1.58411404395506051792494799540, 1.84714489078494559952366711481, 2.08100580542859812053489934032, 2.12010177178148880065823325999, 2.34646427030698464644893233113, 2.47982372484162658418930824924, 2.53001806298188319935418489845, 2.69928626153238513022453264921, 2.82756636959243666788269527181, 2.84961421672070501153407091319, 2.91432149632903365904808842496, 3.41439945873293673447109025019, 3.42656809080920182652120076723, 3.42933872523375038331995076719, 3.59073577326330632765067038176, 3.76660575400644437269900305099
Plot not available for L-functions of degree greater than 10.
