| L(s) = 1 | − 58·7-s − 658·13-s + 7.19e3·19-s − 1.52e3·25-s − 1.04e4·31-s + 3.51e4·37-s − 3.99e4·43-s + 3.44e4·49-s + 2.13e3·61-s + 1.24e5·67-s − 1.92e5·73-s − 9.99e4·79-s + 3.81e4·91-s − 2.58e4·97-s + 1.55e5·103-s − 7.08e4·109-s + 3.14e5·121-s + 127-s + 131-s − 4.17e5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.447·7-s − 1.07·13-s + 4.57·19-s − 0.488·25-s − 1.95·31-s + 4.21·37-s − 3.29·43-s + 2.05·49-s + 0.0735·61-s + 3.37·67-s − 4.22·73-s − 1.80·79-s + 0.483·91-s − 0.278·97-s + 1.43·103-s − 0.571·109-s + 1.95·121-s + 5.50e−6·127-s + 5.09e−6·131-s − 2.04·133-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.455479910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455479910\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 1526 T^{2} - 7436949 T^{4} + 1526 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 29 T - 15966 T^{2} + 29 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 314326 T^{2} + 72863409675 T^{4} - 314326 p^{10} T^{6} + p^{20} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 329 T - 263052 T^{2} + 329 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2020286 T^{2} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 1799 T + p^{5} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 198770 T^{2} - 41387001700749 T^{4} + 198770 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 39031642 T^{2} + 1102761843915963 T^{4} - 39031642 p^{10} T^{6} + p^{20} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 5228 T - 1297167 T^{2} + 5228 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8783 T + p^{5} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 9156974 T^{2} - 13338809137315725 T^{4} + 9156974 p^{10} T^{6} + p^{20} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 19976 T + 252032133 T^{2} + 19976 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 341046910 T^{2} + 63713862584718051 T^{4} - 341046910 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 31068470 T^{2} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 1396994998 T^{2} + 1440478271136378603 T^{4} - 1396994998 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 1069 T - 843453540 T^{2} - 1069 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 62077 T + 2503428822 T^{2} - 62077 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 1457026126 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 48079 T + p^{5} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 49979 T - 579155958 T^{2} + 49979 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 4552161670 T^{2} + 5206134682611335451 T^{4} - 4552161670 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 3438704914 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 12917 T - 8420491368 T^{2} + 12917 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.34669652043767713883976965375, −7.34382004979477732296104575439, −7.18990126672394278072781452705, −7.03069124297558231810007965495, −6.62572848272687237131323087390, −6.00874814335325742718141907941, −5.99093022276243437509658538085, −5.76582560718863177388792798310, −5.37816597507941014548943976947, −5.36413274222223989383758186028, −4.81338017822091862590186409125, −4.80054777104255867474469001166, −4.24242363037715205516459641607, −4.14530909256683027975470334511, −3.43716423012111872291720952693, −3.33989667236541766142447681075, −3.29709568770489529308794944179, −2.76353421511357337076510849333, −2.51855257169075631848876412270, −2.15074753552206696132617107140, −1.67167902106029621960973552367, −1.25546665282057603361865804219, −0.927100071519429410835367512197, −0.74705358850721269413874412768, −0.15194050770754977453500940621,
0.15194050770754977453500940621, 0.74705358850721269413874412768, 0.927100071519429410835367512197, 1.25546665282057603361865804219, 1.67167902106029621960973552367, 2.15074753552206696132617107140, 2.51855257169075631848876412270, 2.76353421511357337076510849333, 3.29709568770489529308794944179, 3.33989667236541766142447681075, 3.43716423012111872291720952693, 4.14530909256683027975470334511, 4.24242363037715205516459641607, 4.80054777104255867474469001166, 4.81338017822091862590186409125, 5.36413274222223989383758186028, 5.37816597507941014548943976947, 5.76582560718863177388792798310, 5.99093022276243437509658538085, 6.00874814335325742718141907941, 6.62572848272687237131323087390, 7.03069124297558231810007965495, 7.18990126672394278072781452705, 7.34382004979477732296104575439, 7.34669652043767713883976965375
