| L(s) = 1 | − 8·7-s − 16·13-s + 48·31-s − 16·43-s + 32·49-s + 48·61-s − 48·67-s − 40·73-s + 128·91-s + 8·97-s − 8·103-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 4.43·13-s + 8.62·31-s − 2.43·43-s + 32/7·49-s + 6.14·61-s − 5.86·67-s − 4.68·73-s + 13.4·91-s + 0.812·97-s − 0.788·103-s + 3.63·121-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 + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3699980099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3699980099\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + 4 T + 8 T^{2} - 4 T^{3} - 62 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 20 T^{2} + 262 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 8 T + 32 T^{2} + 88 T^{3} + 238 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 316 T^{4} + 64006 T^{8} - 316 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 28 T^{2} + 598 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 158 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 80 T^{2} + 2962 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 1358 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 8 T + 32 T^{2} + 88 T^{3} - 782 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 1604 T^{4} + 4130566 T^{8} + 1604 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 180 T^{2} + 14342 T^{4} + 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 24 T + 288 T^{2} + 2376 T^{3} + 18578 T^{4} + 2376 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 13678 T^{4} + 1660 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 76 T^{2} + 5926 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 2404 T^{4} - 43030554 T^{8} + 2404 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 176 T^{2} + 15586 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.96266267944122275292917979590, −4.95006272674043062342521509716, −4.58951536271278105188430941958, −4.56601842448959397473806464142, −4.55095635919019304107887978581, −4.53789091013183312405727055079, −4.41553447727808354651167594360, −4.12758134964691048001470966984, −3.88619116439759465582619811756, −3.72005732737048326520348541771, −3.52186598477977075525965873129, −3.15111566919436405490778005006, −3.10086776286565307561107331628, −2.92657229829980087705000901642, −2.84444473672419754934912943438, −2.81504521716507621262939527475, −2.70535345296119346195760227438, −2.42304487944256589698408059732, −2.33379078133200493294832038044, −2.02681507606088589201715121493, −1.70003351839118246004774963364, −1.37278999509265158877939824350, −0.793328701528245584417477695339, −0.77958564745287889848075972126, −0.19192887249395921764545045393,
0.19192887249395921764545045393, 0.77958564745287889848075972126, 0.793328701528245584417477695339, 1.37278999509265158877939824350, 1.70003351839118246004774963364, 2.02681507606088589201715121493, 2.33379078133200493294832038044, 2.42304487944256589698408059732, 2.70535345296119346195760227438, 2.81504521716507621262939527475, 2.84444473672419754934912943438, 2.92657229829980087705000901642, 3.10086776286565307561107331628, 3.15111566919436405490778005006, 3.52186598477977075525965873129, 3.72005732737048326520348541771, 3.88619116439759465582619811756, 4.12758134964691048001470966984, 4.41553447727808354651167594360, 4.53789091013183312405727055079, 4.55095635919019304107887978581, 4.56601842448959397473806464142, 4.58951536271278105188430941958, 4.95006272674043062342521509716, 4.96266267944122275292917979590
Plot not available for L-functions of degree greater than 10.
