| L(s) = 1 | − 8·31-s − 56·61-s + 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 1.43·31-s − 7.17·61-s + 1.45·121-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 + 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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.009762727785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.009762727785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 23 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 337 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 206 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 542 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + T + p T^{2} )^{8} \) |
| 37 | \( ( 1 + 1106 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 3191 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 2818 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 5518 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 7 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 2471 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 7298 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 11822 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 16609 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−3.49014574080605191416102204361, −3.45954302306274973803897772413, −3.28447363031352732101139602363, −3.21344631506859489464536598237, −3.07558765993966612196336496705, −2.94292795333945612394149840574, −2.83040695455142206453226635162, −2.62536474830643191962765503279, −2.56908557195509899418616543068, −2.56466152726544929201166502995, −2.56152219494541457428512649476, −2.04891307316205908245486351561, −2.03434313028357132343124398147, −1.88610740943763290888904766175, −1.80677839617563531066721894198, −1.67993914202337495858294490719, −1.64638992772055749020819110480, −1.44740064273814247445980590420, −1.23206700267576099070560714358, −1.04928644689427286756683753151, −0.944650187849310353576273854521, −0.874781380476969319450207383157, −0.41770995523511804752814332521, −0.26549009970040889200333098656, −0.01159600796646830542793910022,
0.01159600796646830542793910022, 0.26549009970040889200333098656, 0.41770995523511804752814332521, 0.874781380476969319450207383157, 0.944650187849310353576273854521, 1.04928644689427286756683753151, 1.23206700267576099070560714358, 1.44740064273814247445980590420, 1.64638992772055749020819110480, 1.67993914202337495858294490719, 1.80677839617563531066721894198, 1.88610740943763290888904766175, 2.03434313028357132343124398147, 2.04891307316205908245486351561, 2.56152219494541457428512649476, 2.56466152726544929201166502995, 2.56908557195509899418616543068, 2.62536474830643191962765503279, 2.83040695455142206453226635162, 2.94292795333945612394149840574, 3.07558765993966612196336496705, 3.21344631506859489464536598237, 3.28447363031352732101139602363, 3.45954302306274973803897772413, 3.49014574080605191416102204361
Plot not available for L-functions of degree greater than 10.
