| L(s) = 1 | − 168·17-s + 352·25-s − 24·41-s − 1.23e3·49-s + 304·73-s − 6.19e3·89-s + 3.22e3·97-s − 8.13e3·113-s + 3.70e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.34e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.39·17-s + 2.81·25-s − 0.0914·41-s − 3.59·49-s + 0.487·73-s − 7.37·89-s + 3.37·97-s − 6.77·113-s + 2.77·121-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 + 6.13·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.116516007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.116516007\) |
| \(L(\frac{5}{2})\) |
|
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 | \( ( 1 - 176 T^{2} + 24657 T^{4} - 176 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( ( 1 + 88 p T^{2} + 315825 T^{4} + 88 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 1850 T^{2} + 3481179 T^{4} - 1850 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 6736 T^{2} + 20939694 T^{4} - 6736 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 + 42 T + 8674 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 19 | \( ( 1 + 4936 T^{2} - 5853666 T^{4} + 4936 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 + 5576 T^{2} + 90452814 T^{4} + 5576 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 31136 T^{2} + 1232380878 T^{4} - 31136 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 18616 T^{2} + 1461801633 T^{4} + 18616 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 35968 T^{2} - 1450338258 T^{4} - 35968 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T + 98026 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 - 180860 T^{2} + 18242862486 T^{4} - 180860 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 + 285692 T^{2} + 39669490374 T^{4} + 285692 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 523256 T^{2} + 112424539929 T^{4} - 523256 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 490748 T^{2} + 140692742358 T^{4} - 490748 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 21460 T^{2} + 92453368470 T^{4} - 21460 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 601126 T^{2} + p^{6} T^{4} )^{4} \) |
| 71 | \( ( 1 + 1252472 T^{2} + 640981027950 T^{4} + 1252472 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 76 T + 650445 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 79 | \( ( 1 + 1918912 T^{2} + 1406425119486 T^{4} + 1918912 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 - 1820426 T^{2} + 1469807863755 T^{4} - 1820426 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 + 1548 T + 1492882 T^{2} + 1548 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 - 806 T + 1758363 T^{2} - 806 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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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
−3.45283416785013554837606682201, −3.35574997634629955482822032451, −3.27971440419066121235712220382, −3.25155509657697070110463879198, −3.09856868487461781085138774510, −2.90749452878888656753084889767, −2.65517626130723907736959311923, −2.51761089074934207145791874072, −2.50508329099888485457319960600, −2.44887448492930588323585939525, −2.37802664525908101411225911726, −2.10876787636461743366480871122, −2.05347344487593220017397850360, −1.68189001009827484078709314313, −1.59773549971785956342591427238, −1.55664934802937882475457124265, −1.30041657341215258899885615929, −1.21353645359614542259784971800, −1.17482655371208842310231851303, −1.15328432784931362057166838428, −0.58591660366425965231291782859, −0.47624660734728009801307157406, −0.37248752750392918653529003846, −0.33084168405482716410499986269, −0.13593975018673752595223522197,
0.13593975018673752595223522197, 0.33084168405482716410499986269, 0.37248752750392918653529003846, 0.47624660734728009801307157406, 0.58591660366425965231291782859, 1.15328432784931362057166838428, 1.17482655371208842310231851303, 1.21353645359614542259784971800, 1.30041657341215258899885615929, 1.55664934802937882475457124265, 1.59773549971785956342591427238, 1.68189001009827484078709314313, 2.05347344487593220017397850360, 2.10876787636461743366480871122, 2.37802664525908101411225911726, 2.44887448492930588323585939525, 2.50508329099888485457319960600, 2.51761089074934207145791874072, 2.65517626130723907736959311923, 2.90749452878888656753084889767, 3.09856868487461781085138774510, 3.25155509657697070110463879198, 3.27971440419066121235712220382, 3.35574997634629955482822032451, 3.45283416785013554837606682201
Plot not available for L-functions of degree greater than 10.
