| L(s) = 1 | − 6·7-s − 14·13-s + 76·19-s − 42·25-s − 20·31-s + 252·37-s − 100·43-s + 107·49-s − 158·61-s + 154·67-s − 68·73-s − 22·79-s + 84·91-s + 194·97-s + 266·103-s + 520·109-s + 150·121-s + 127-s + 131-s − 456·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 6/7·7-s − 1.07·13-s + 4·19-s − 1.67·25-s − 0.645·31-s + 6.81·37-s − 2.32·43-s + 2.18·49-s − 2.59·61-s + 2.29·67-s − 0.931·73-s − 0.278·79-s + 0.923·91-s + 2·97-s + 2.58·103-s + 4.77·109-s + 1.23·121-s + 0.00787·127-s + 0.00763·131-s − 3.42·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.909566942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.909566942\) |
| \(L(2)\) |
|
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 | $C_2^3$ | \( 1 + 42 T^{2} + 1139 T^{4} + 42 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 3 T - 40 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 150 T^{2} + 7859 T^{4} - 150 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 7 T - 120 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 378 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 1050 T^{2} + 822659 T^{4} + 1050 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 910 T^{2} + 120819 T^{4} - 910 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T - 861 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 63 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 770 T^{2} - 2232861 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 50 T + 651 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 2618 T^{2} + 1974243 T^{4} + 2618 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 2838 T^{2} - 4063117 T^{4} - 2838 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 79 T + 2520 T^{2} + 79 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 77 T + 1440 T^{2} - 77 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 3810 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 12210 T^{2} + 101625779 T^{4} + 12210 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 14042 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{4}( 1 + p T + p^{2} T^{2} )^{2} \) |
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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
−6.87037783945823062431972496022, −6.14704933806061362320756842291, −6.12565439742956953229869616258, −6.10654666418732378222897544232, −6.04252520665429107263095169123, −5.53492411753573169997552400517, −5.42353637778834402117918997794, −5.23443783562327482217348583923, −4.91232889063056775073166382696, −4.55325688903701862554080738197, −4.39683739442502078387847496932, −4.35360711470117415172516376309, −4.00421182100879090870599369887, −3.53595154328225787872280760512, −3.27421622236035220393855032146, −3.13969319145737989898924138877, −3.01511724546377481279346712128, −2.83196728647583328906167186185, −2.18017770991185643217132816827, −2.05560021116344158318509884036, −1.95256062285156470670450291235, −1.20947959691385725252870370409, −0.825710898761421900616034597873, −0.70435492150494376779593283097, −0.48002978946361755208326381191,
0.48002978946361755208326381191, 0.70435492150494376779593283097, 0.825710898761421900616034597873, 1.20947959691385725252870370409, 1.95256062285156470670450291235, 2.05560021116344158318509884036, 2.18017770991185643217132816827, 2.83196728647583328906167186185, 3.01511724546377481279346712128, 3.13969319145737989898924138877, 3.27421622236035220393855032146, 3.53595154328225787872280760512, 4.00421182100879090870599369887, 4.35360711470117415172516376309, 4.39683739442502078387847496932, 4.55325688903701862554080738197, 4.91232889063056775073166382696, 5.23443783562327482217348583923, 5.42353637778834402117918997794, 5.53492411753573169997552400517, 6.04252520665429107263095169123, 6.10654666418732378222897544232, 6.12565439742956953229869616258, 6.14704933806061362320756842291, 6.87037783945823062431972496022
