| L(s) = 1 | + 284·25-s + 940·49-s − 1.28e3·73-s + 2.29e3·97-s + 5.26e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.75e4·169-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 + ⋯ |
| L(s) = 1 | + 2.27·25-s + 2.74·49-s − 2.06·73-s + 2.40·97-s + 3.95·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 8·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(14.02114387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.02114387\) |
| \(L(\frac{5}{2})\) |
|
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 | \( ( 1 - 142 T^{2} + 4539 T^{4} - 142 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( ( 1 - 470 T^{2} + 103251 T^{4} - 470 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 126 T + 6623 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} )^{2}( 1 + 126 T + 6623 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 17 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 19 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 23 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 29 | \( ( 1 + 1150 T^{2} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 + 54682 T^{2} + 2102617443 T^{4} + 54682 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 41 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 43 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 47 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 53 | \( ( 1 - 38446 T^{2} - 20686266213 T^{4} - 38446 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 + 103430 T^{2} + p^{6} T^{4} )^{4} \) |
| 61 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 67 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 71 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 73 | \( ( 1 + 322 T - 285333 T^{2} + 322 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 79 | \( ( 1 + 890822 T^{2} + p^{6} T^{4} )^{4} \) |
| 83 | \( ( 1 - 1530 T + 1352087 T^{2} - 1530 p^{3} T^{3} + p^{6} T^{4} )^{2}( 1 + 1530 T + 1352087 T^{2} + 1530 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 97 | \( ( 1 - 574 T - 583197 T^{2} - 574 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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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
−3.53771023620112635753012448274, −3.47191074380228255829273700163, −3.36731455993187816006000070711, −3.00608745918441258100371262923, −3.00537923533238538888345473837, −2.96302846257440436426636722083, −2.80710971356938535959448214209, −2.55518823318183839013848617267, −2.54646805351729937467679154065, −2.45621567536351298569750852270, −2.26795281373470640365500841564, −1.97983552459157085337686332843, −1.91250487331663417477359318392, −1.91124110912727304608270272615, −1.80250136947160897355927547856, −1.47240105764935653483230976916, −1.30513344451521994205020323121, −1.16715199747438864154852866651, −1.10662528143205462474768873094, −0.882308007455229233761018759475, −0.71050355990875859017752679713, −0.67501111275389604850478913148, −0.46315754505893177143653726586, −0.28145028907757604058026380581, −0.18511596569642236505261632756,
0.18511596569642236505261632756, 0.28145028907757604058026380581, 0.46315754505893177143653726586, 0.67501111275389604850478913148, 0.71050355990875859017752679713, 0.882308007455229233761018759475, 1.10662528143205462474768873094, 1.16715199747438864154852866651, 1.30513344451521994205020323121, 1.47240105764935653483230976916, 1.80250136947160897355927547856, 1.91124110912727304608270272615, 1.91250487331663417477359318392, 1.97983552459157085337686332843, 2.26795281373470640365500841564, 2.45621567536351298569750852270, 2.54646805351729937467679154065, 2.55518823318183839013848617267, 2.80710971356938535959448214209, 2.96302846257440436426636722083, 3.00537923533238538888345473837, 3.00608745918441258100371262923, 3.36731455993187816006000070711, 3.47191074380228255829273700163, 3.53771023620112635753012448274
Plot not available for L-functions of degree greater than 10.
