| L(s) = 1 | + 4-s − 8·13-s + 4·16-s − 10·25-s − 32·37-s + 2·49-s − 8·52-s + 16·61-s + 11·64-s + 40·73-s − 44·97-s − 10·100-s − 32·109-s + 38·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 76·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.21·13-s + 16-s − 2·25-s − 5.26·37-s + 2/7·49-s − 1.10·52-s + 2.04·61-s + 11/8·64-s + 4.68·73-s − 4.46·97-s − 100-s − 3.06·109-s + 3.45·121-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 + 0.0773·167-s + 5.84·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9300756559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9300756559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 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 | \( ( 1 - 19 T^{2} + 240 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 5 T + p T^{2} )^{4}( 1 + 7 T + p T^{2} )^{4} \) |
| 17 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - p T^{2} )^{8} \) |
| 23 | \( ( 1 + 2 T^{2} - 525 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 38 T^{2} + 603 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 47 T^{2} + 1248 T^{4} + 47 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 2 T^{2} - 1677 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 26 T^{2} - 1173 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 82 T^{2} + 4515 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 101 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 74 T^{2} + 987 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 5 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 - 19 T^{2} - 6528 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−5.23150967764624715764226567605, −5.07647154848516363188659078678, −5.04608334174836024508975987585, −4.79441689485498948599671450926, −4.56506110634631070314352158379, −4.48501086374034286918113710401, −4.47699873582168140844378123716, −3.89852525519542897527722209064, −3.89809402654919054630632816367, −3.71860273309136010237923747477, −3.64059817601968193976126765481, −3.60840385404720358532507287784, −3.41479695010267861652246598755, −3.29340296443044590444916737323, −2.70486903593500102404765870027, −2.61971195312539969638161056619, −2.56595349773541880634582309967, −2.55112122771473215818551170326, −2.12802550718790394432427373894, −2.00341630920871864525283406034, −1.72321818550642448142478171157, −1.47403551636966006944133902116, −1.43637849241101149154827117920, −0.72037794184621274216933273892, −0.25698170841166324695828238336,
0.25698170841166324695828238336, 0.72037794184621274216933273892, 1.43637849241101149154827117920, 1.47403551636966006944133902116, 1.72321818550642448142478171157, 2.00341630920871864525283406034, 2.12802550718790394432427373894, 2.55112122771473215818551170326, 2.56595349773541880634582309967, 2.61971195312539969638161056619, 2.70486903593500102404765870027, 3.29340296443044590444916737323, 3.41479695010267861652246598755, 3.60840385404720358532507287784, 3.64059817601968193976126765481, 3.71860273309136010237923747477, 3.89809402654919054630632816367, 3.89852525519542897527722209064, 4.47699873582168140844378123716, 4.48501086374034286918113710401, 4.56506110634631070314352158379, 4.79441689485498948599671450926, 5.04608334174836024508975987585, 5.07647154848516363188659078678, 5.23150967764624715764226567605
Plot not available for L-functions of degree greater than 10.
