| L(s) = 1 | + 8·7-s − 2·16-s + 8·31-s + 4·49-s + 16·73-s − 16·79-s + 8·97-s + 16·103-s − 16·112-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 1/2·16-s + 1.43·31-s + 4/7·49-s + 1.87·73-s − 1.80·79-s + 0.812·97-s + 1.57·103-s − 1.51·112-s + 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.870202076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.870202076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T^{4} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 12 T^{2} + 62 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 20 T^{2} + 462 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 26 T^{2} + 291 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 60 T^{2} + 1742 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 68 T^{2} + 2622 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 84 T^{2} + 4526 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 122 T^{2} + 6819 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 60 T^{2} + 3782 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 132 T^{2} + 9374 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 108 T^{2} + 8342 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 134 T^{2} + 9531 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 58 T^{2} + 4419 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 36 T^{2} + 9806 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 140 T^{2} + 15222 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 36 T^{2} + 6566 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 2 T - 21 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.04105363443865901457806474467, −3.89115712949937940251117037257, −3.53400170288551723588177435102, −3.48955730769732076557401715876, −3.47055898013377239928161182779, −3.35934275328586753029241658830, −3.14825515335034078899782942903, −2.93261480875675909946970908441, −2.83820331743864069177256214239, −2.83765552097132693693815197951, −2.57165191350672775133498316551, −2.43942578785479028475393176680, −2.37929419790749836720766596997, −2.07631711600582465096000456574, −2.00811906353746893354011704197, −1.88327888920698941802580306510, −1.79621132344828826128213495041, −1.60617159800932713668654288670, −1.37823917604624685472113553872, −1.30714413345498410540912184368, −1.25566580934427710848524540412, −0.907630878517414812700411563455, −0.72928770815477545565338199301, −0.47650774105457580312866090495, −0.18597290509317333110233525168,
0.18597290509317333110233525168, 0.47650774105457580312866090495, 0.72928770815477545565338199301, 0.907630878517414812700411563455, 1.25566580934427710848524540412, 1.30714413345498410540912184368, 1.37823917604624685472113553872, 1.60617159800932713668654288670, 1.79621132344828826128213495041, 1.88327888920698941802580306510, 2.00811906353746893354011704197, 2.07631711600582465096000456574, 2.37929419790749836720766596997, 2.43942578785479028475393176680, 2.57165191350672775133498316551, 2.83765552097132693693815197951, 2.83820331743864069177256214239, 2.93261480875675909946970908441, 3.14825515335034078899782942903, 3.35934275328586753029241658830, 3.47055898013377239928161182779, 3.48955730769732076557401715876, 3.53400170288551723588177435102, 3.89115712949937940251117037257, 4.04105363443865901457806474467
Plot not available for L-functions of degree greater than 10.
