| L(s) = 1 | + 2·4-s + 16-s + 24·19-s + 2·25-s + 12·31-s − 16·37-s + 32·43-s − 24·61-s − 2·64-s + 40·67-s − 24·73-s + 48·76-s + 28·79-s + 4·100-s + 72·103-s − 40·109-s − 26·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 4-s + 1/4·16-s + 5.50·19-s + 2/5·25-s + 2.15·31-s − 2.63·37-s + 4.87·43-s − 3.07·61-s − 1/4·64-s + 4.88·67-s − 2.80·73-s + 5.50·76-s + 3.15·79-s + 2/5·100-s + 7.09·103-s − 3.83·109-s − 2.36·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.48208856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.48208856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( 1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 1024 p^{2} T^{10} + 478 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 12 T + 92 T^{2} - 528 T^{3} + 2487 T^{4} - 528 p T^{5} + 92 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 28 T^{2} + 255 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 62 T^{2} + 1995 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 6 T + 23 T^{2} - 66 T^{3} - 468 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 8 T - 8 T^{2} - 16 T^{3} + 1447 T^{4} - 16 p T^{5} - 8 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 20 T^{2} - 1146 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 8 T + 84 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 - 152 T^{2} + 13198 T^{4} - 834176 T^{6} + 42212419 T^{8} - 834176 p^{2} T^{10} + 13198 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 883694 p^{2} T^{10} + 13753 p^{4} T^{12} + 158 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 30058 p^{2} T^{10} - 4727 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 12 T + 176 T^{2} + 1536 T^{3} + 15591 T^{4} + 1536 p T^{5} + 176 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 176 T^{2} + 15234 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 16131 T^{4} + 1608 p T^{5} + 182 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 14 T + 7 T^{2} - 434 T^{3} + 13996 T^{4} - 434 p T^{5} + 7 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 278 T^{2} + 32811 T^{4} + 278 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 190 T^{2} + 20643 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.29845765106951543254187959529, −4.14356543546953991728425655985, −4.08386710965034839805966612249, −3.99180190379906857045287788220, −3.87337227124576847480095321658, −3.57609637638814789604612707075, −3.51565483152986228017917148192, −3.28977078279782274063594178792, −3.27004519704467688194320881899, −3.12407429221355645177440607439, −3.02568056876845092854215626503, −2.86980643062670198621531290564, −2.69395525088350272172340010124, −2.52317465397626077753882305924, −2.45443090521383463091287558454, −2.37428352175997542567851884347, −1.91215397375795434372622134121, −1.78468534204426587935879380347, −1.72786654934789915777283165123, −1.62194154475146910129968374014, −1.05983981774800557796664473423, −0.960530131540207451571737132879, −0.921958499226523987839982984909, −0.849700686797922314823507543970, −0.42903176129540531240919976385,
0.42903176129540531240919976385, 0.849700686797922314823507543970, 0.921958499226523987839982984909, 0.960530131540207451571737132879, 1.05983981774800557796664473423, 1.62194154475146910129968374014, 1.72786654934789915777283165123, 1.78468534204426587935879380347, 1.91215397375795434372622134121, 2.37428352175997542567851884347, 2.45443090521383463091287558454, 2.52317465397626077753882305924, 2.69395525088350272172340010124, 2.86980643062670198621531290564, 3.02568056876845092854215626503, 3.12407429221355645177440607439, 3.27004519704467688194320881899, 3.28977078279782274063594178792, 3.51565483152986228017917148192, 3.57609637638814789604612707075, 3.87337227124576847480095321658, 3.99180190379906857045287788220, 4.08386710965034839805966612249, 4.14356543546953991728425655985, 4.29845765106951543254187959529
Plot not available for L-functions of degree greater than 10.
