| L(s) = 1 | + 2.01e3·13-s + 5.04e4·25-s − 1.51e5·37-s − 3.46e5·49-s − 1.17e6·61-s − 1.55e6·73-s − 6.10e6·97-s − 3.10e6·109-s − 5.10e6·121-s + ⋯ |
| L(s) = 1 | + 0.917·13-s + 3.22·25-s − 2.98·37-s − 2.94·49-s − 5.17·61-s − 4.00·73-s − 6.69·97-s − 2.39·109-s − 2.88·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.9054571636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9054571636\) |
| \(L(4)\) |
|
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 | $C_2^2$ | \( ( 1 - 1008 p^{2} T^{2} + p^{12} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 173090 T^{2} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2553262 T^{2} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 504 T + p^{6} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 40584096 T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 45320690 T^{2} + p^{12} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 143662178 T^{2} + p^{12} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 735738192 T^{2} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1747573634 T^{2} + p^{12} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 37802 T + p^{6} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 8808809184 T^{2} + p^{12} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 925476050 T^{2} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 21259707842 T^{2} + p^{12} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 42319322640 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 60926567186 T^{2} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 293830 T + p^{6} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 41321763506 T^{2} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 230443345442 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 389088 T + p^{6} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 432553205950 T^{2} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 412457843954 T^{2} + p^{12} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 143152027200 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1527120 T + p^{6} T^{2} )^{4} \) |
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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
−7.37153597072835220154746664156, −7.20693820048018447453837064223, −7.16518951625974573401350507926, −6.57361924283853613123999451882, −6.42618282639450295330305154510, −6.25593834765884811842280576165, −6.22373268624511783902319928684, −5.37268557147421247559273182966, −5.36954809445636766944610491868, −5.17971505726378626685593115992, −4.95848109981560908742437083201, −4.35666528653930050989086235047, −4.26918082524338469569436589268, −4.11702269327345479956554815735, −3.52334597534757930654648222734, −3.07939804695267762585435534663, −2.99327620652935684483189599455, −2.92786326738147928127939690100, −2.53391856584142563219518741199, −1.57824732177666639133303481221, −1.54145404708273497165130253055, −1.44198315642500891293536423798, −1.24110426086958319680851422625, −0.36247485584408369126209976959, −0.16125032756029241918581659310,
0.16125032756029241918581659310, 0.36247485584408369126209976959, 1.24110426086958319680851422625, 1.44198315642500891293536423798, 1.54145404708273497165130253055, 1.57824732177666639133303481221, 2.53391856584142563219518741199, 2.92786326738147928127939690100, 2.99327620652935684483189599455, 3.07939804695267762585435534663, 3.52334597534757930654648222734, 4.11702269327345479956554815735, 4.26918082524338469569436589268, 4.35666528653930050989086235047, 4.95848109981560908742437083201, 5.17971505726378626685593115992, 5.36954809445636766944610491868, 5.37268557147421247559273182966, 6.22373268624511783902319928684, 6.25593834765884811842280576165, 6.42618282639450295330305154510, 6.57361924283853613123999451882, 7.16518951625974573401350507926, 7.20693820048018447453837064223, 7.37153597072835220154746664156
