| L(s) = 1 | + 2·25-s − 8·29-s + 4·41-s − 4·53-s − 4·73-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 + ⋯ |
| L(s) = 1 | + 2·25-s − 8·29-s + 4·41-s − 4·53-s − 4·73-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7450181426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7450181426\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T^{4} + T^{8} \) |
| 17 | \( ( 1 + T^{2} )^{4} \) |
| good | 3 | \( 1 - T^{8} + T^{16} \) |
| 5 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 7 | \( 1 - T^{8} + T^{16} \) |
| 11 | \( 1 - T^{8} + T^{16} \) |
| 19 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 23 | \( 1 - T^{8} + T^{16} \) |
| 29 | \( ( 1 + T )^{8}( 1 - T^{4} + T^{8} ) \) |
| 31 | \( ( 1 + T^{8} )^{2} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 + T^{4} )^{2} \) |
| 41 | \( ( 1 - T + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 43 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{2} )^{8} \) |
| 53 | \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 + T^{4} )^{2} \) |
| 67 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 71 | \( 1 - T^{8} + T^{16} \) |
| 73 | \( ( 1 + T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 79 | \( ( 1 + T^{8} )^{2} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 97 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
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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
−3.83604199787752130013235643880, −3.46098726582716841106992669200, −3.32489814753124103396545118388, −3.31150733472183035030016546397, −3.26901620750522454862907677715, −3.25922430005267260929103625131, −3.20974008903981208177965894576, −3.17159145744035666423022659633, −2.76261172933659029915187055230, −2.68573255567198705820496670675, −2.68297601521050924238559347877, −2.41807508729575740151812500095, −2.16969974153752866388163387404, −2.09303047944947960745987061120, −2.07267781265642578310200451326, −1.90473692936158133708360767864, −1.89887072382032970389084979477, −1.83742370597555535591924669022, −1.40212629615332874493893671279, −1.30451498561095344976216545250, −1.30180530434356063640986682518, −1.15212896993315467230500717658, −0.886442874829404461433483742328, −0.48314063890882040608677459928, −0.26993801071272039359165167898,
0.26993801071272039359165167898, 0.48314063890882040608677459928, 0.886442874829404461433483742328, 1.15212896993315467230500717658, 1.30180530434356063640986682518, 1.30451498561095344976216545250, 1.40212629615332874493893671279, 1.83742370597555535591924669022, 1.89887072382032970389084979477, 1.90473692936158133708360767864, 2.07267781265642578310200451326, 2.09303047944947960745987061120, 2.16969974153752866388163387404, 2.41807508729575740151812500095, 2.68297601521050924238559347877, 2.68573255567198705820496670675, 2.76261172933659029915187055230, 3.17159145744035666423022659633, 3.20974008903981208177965894576, 3.25922430005267260929103625131, 3.26901620750522454862907677715, 3.31150733472183035030016546397, 3.32489814753124103396545118388, 3.46098726582716841106992669200, 3.83604199787752130013235643880
Plot not available for L-functions of degree greater than 10.
