| L(s) = 1 | + 10·25-s + 36·41-s − 2·49-s + 64·73-s + 96·89-s − 4·97-s + 12·113-s − 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 2·25-s + 5.62·41-s − 2/7·49-s + 7.49·73-s + 10.1·89-s − 0.406·97-s + 1.12·113-s − 2.36·121-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 − 1.69·169-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 + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4452529887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4452529887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + T^{2} - 48 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 11 T^{2} - 48 T^{4} + 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 31 T^{2} + 432 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 43 T^{2} + 1008 T^{4} + 43 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 47 T^{2} + 1248 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 37 T^{2} - 480 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 79 T^{2} + 4032 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 109 T^{2} + 8400 T^{4} + 109 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 13 T^{2} - 3552 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 143 T^{2} + 14208 T^{4} - 143 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 157 T^{2} + 17760 T^{4} + 157 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 12 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.91067262517487564503949347359, −3.70849048308428593812936501319, −3.62658144849637882857276457357, −3.50578539339254336526733859257, −3.45153655713605095835442867697, −3.36440804851288505089115091758, −3.19306582013311400009082413617, −3.08522870401622026305255721537, −3.05944668302515222101841018971, −2.79601580276702422997081623487, −2.39144527626817456223326261830, −2.38950210778554814943273164673, −2.25028152991399401013664697368, −2.23024440273012056275551189992, −2.21558523466826844170685379738, −2.20014901839750596189794580487, −1.87969480673275856951826264230, −1.72621938999425751733537117767, −1.20846267383728973074569124054, −1.08427281690167993489433236308, −0.931299665593158425479742861253, −0.857274569487811462988033166934, −0.800997812469302089562501330940, −0.78642157028277398877706087950, −0.04633864865943620649984461345,
0.04633864865943620649984461345, 0.78642157028277398877706087950, 0.800997812469302089562501330940, 0.857274569487811462988033166934, 0.931299665593158425479742861253, 1.08427281690167993489433236308, 1.20846267383728973074569124054, 1.72621938999425751733537117767, 1.87969480673275856951826264230, 2.20014901839750596189794580487, 2.21558523466826844170685379738, 2.23024440273012056275551189992, 2.25028152991399401013664697368, 2.38950210778554814943273164673, 2.39144527626817456223326261830, 2.79601580276702422997081623487, 3.05944668302515222101841018971, 3.08522870401622026305255721537, 3.19306582013311400009082413617, 3.36440804851288505089115091758, 3.45153655713605095835442867697, 3.50578539339254336526733859257, 3.62658144849637882857276457357, 3.70849048308428593812936501319, 3.91067262517487564503949347359
Plot not available for L-functions of degree greater than 10.
