| L(s) = 1 | + 95·4-s + 675·9-s − 1.20e3·11-s + 4.73e3·16-s − 1.26e3·19-s − 1.17e4·29-s − 792·31-s + 6.41e4·36-s + 3.54e3·41-s − 1.14e5·44-s − 4.80e3·49-s − 1.19e5·59-s − 1.03e5·61-s + 1.39e5·64-s + 1.61e5·71-s − 1.19e5·76-s + 1.03e5·79-s + 2.24e5·81-s + 7.53e4·89-s − 8.11e5·99-s + 1.58e5·101-s + 4.98e5·109-s − 1.11e6·116-s + 5.64e5·121-s − 7.52e4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2.96·4-s + 25/9·9-s − 2.99·11-s + 4.62·16-s − 0.800·19-s − 2.59·29-s − 0.148·31-s + 8.24·36-s + 0.329·41-s − 8.89·44-s − 2/7·49-s − 4.45·59-s − 3.56·61-s + 4.27·64-s + 3.80·71-s − 2.37·76-s + 1.86·79-s + 3.80·81-s + 1.00·89-s − 8.31·99-s + 1.54·101-s + 4.02·109-s − 7.71·116-s + 3.50·121-s − 0.439·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(10.23481371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.23481371\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 95 T^{2} + 67 p^{6} T^{4} - 95 p^{10} T^{6} + p^{20} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 25 p^{3} T^{2} + 2848 p^{4} T^{4} - 25 p^{13} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 601 T + 259506 T^{2} + 601 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1227415 T^{2} + 637156918888 T^{4} - 1227415 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3631055 T^{2} + 7326406953088 T^{4} - 3631055 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 630 T + 4931238 T^{2} + 630 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 794440 T^{2} + 80427886013038 T^{4} + 794440 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 5885 T + 35168948 T^{2} + 5885 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 396 T + 43423646 T^{2} + 396 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 186138220 T^{2} + 16233691530051798 T^{4} - 186138220 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 1774 T - 4379094 T^{2} - 1774 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 192604600 T^{2} + 42335578678671838 T^{4} - 192604600 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 610148955 T^{2} + 179999270966185448 T^{4} - 610148955 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 121403840 T^{2} + 343112212052927758 T^{4} - 121403840 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 59600 T + 1881188438 T^{2} + 59600 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 51846 T + 1832119946 T^{2} + 51846 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 3694295980 T^{2} + 6360450835846875958 T^{4} - 3694295980 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 80744 T + 4781205326 T^{2} - 80744 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 7629610780 T^{2} + 23095701672517150438 T^{4} - 7629610780 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 51795 T + 1771153398 T^{2} - 51795 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9224640060 T^{2} + 49287451417686473558 T^{4} - 9224640060 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 37650 T + 10990453658 T^{2} - 37650 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 23390523975 T^{2} + \)\(25\!\cdots\!08\)\( T^{4} - 23390523975 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−8.069338225052385317957791620713, −7.937993507026858251609741351480, −7.78324681559379128902718192488, −7.37695836599371731146134605929, −7.20948806183889104988766593049, −7.06151764130658653886840182772, −7.04378935092252243847890706520, −6.34524284210113474762338176343, −6.11532737770200459150185844173, −5.96896880584946606066680212604, −5.78100939040502500919961371548, −5.18842814626403583143181418779, −4.76400717139725706658022006417, −4.60804313200843778532823171858, −4.45210214488369582362756230663, −3.51894398908520082493430558655, −3.51002485384486767037220434709, −3.04449121158761777072329511083, −2.77483771216783465775175960640, −2.11283251759477412050170426700, −1.93633797593448588955457164582, −1.83226387416899574484958251428, −1.65015025618872145779883879703, −0.72436155922651881703048570535, −0.40208944417240909277229837993,
0.40208944417240909277229837993, 0.72436155922651881703048570535, 1.65015025618872145779883879703, 1.83226387416899574484958251428, 1.93633797593448588955457164582, 2.11283251759477412050170426700, 2.77483771216783465775175960640, 3.04449121158761777072329511083, 3.51002485384486767037220434709, 3.51894398908520082493430558655, 4.45210214488369582362756230663, 4.60804313200843778532823171858, 4.76400717139725706658022006417, 5.18842814626403583143181418779, 5.78100939040502500919961371548, 5.96896880584946606066680212604, 6.11532737770200459150185844173, 6.34524284210113474762338176343, 7.04378935092252243847890706520, 7.06151764130658653886840182772, 7.20948806183889104988766593049, 7.37695836599371731146134605929, 7.78324681559379128902718192488, 7.937993507026858251609741351480, 8.069338225052385317957791620713
