| L(s) = 1 | + 8·9-s + 104·25-s − 112·29-s + 208·37-s + 4·49-s − 240·53-s + 84·81-s − 624·109-s − 304·113-s − 472·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 936·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 8/9·9-s + 4.15·25-s − 3.86·29-s + 5.62·37-s + 4/49·49-s − 4.52·53-s + 1.03·81-s − 5.72·109-s − 2.69·113-s − 3.90·121-s + 0.00787·127-s + 0.00763·131-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 + 5.53·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.290852968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.290852968\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 4 T^{2} + 38 p^{2} T^{4} - 4 p^{4} T^{6} + p^{8} T^{8} \) |
| good | 3 | \( ( 1 - 4 T^{2} - 2 p^{2} T^{4} - 4 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 5 | \( ( 1 - 52 T^{2} + 1742 T^{4} - 52 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 236 T^{2} + 31430 T^{4} + 236 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 36 p T^{2} + 102862 T^{4} - 36 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 4 T^{2} + 155270 T^{4} - 4 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 1188 T^{2} + 613294 T^{4} - 1188 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 1676 T^{2} + 1214822 T^{4} + 1676 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 28 T + 1142 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 2180 T^{2} + 2458118 T^{4} - 2180 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 52 T + 2678 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 4868 T^{2} + 10725062 T^{4} - 4868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 2540 T^{2} + 7272902 T^{4} + 2540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 6788 T^{2} + 21266822 T^{4} - 6788 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 30 T + p^{2} T^{2} )^{8} \) |
| 59 | \( ( 1 - 4580 T^{2} + 29037038 T^{4} - 4580 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4212 T^{2} + 3996814 T^{4} - 4212 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8150 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 + 11964 T^{2} + 70047686 T^{4} + 11964 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8388 T^{2} + 67882822 T^{4} - 8388 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 1028 T^{2} + 37863942 T^{4} - 1028 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 8132 T^{2} + 106317422 T^{4} - 8132 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 30916 T^{2} + 364289990 T^{4} - 30916 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 19204 T^{2} + 268303110 T^{4} - 19204 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−5.34000673346758254706633579584, −5.16595645460576386564824575078, −4.89024839856780463284394815877, −4.83810840398826483059235908956, −4.58526595759162207004122506029, −4.43331353546425010104741425265, −4.41425959121698848034570837368, −4.18567303099730619734674541181, −4.06358808103318717592658499857, −3.88906274554816288996551382748, −3.80420662922011872959813061205, −3.23729871471545297064946763680, −3.21290243326620881805651351412, −3.12785674896288581037385834987, −2.91068982443971425012796922367, −2.69301788139933848461388439477, −2.68774532603071882282965502138, −2.28694975553724960250194204790, −1.83236468482907489337941901580, −1.76752898517719469551970079677, −1.71969344331057866791216969655, −1.07482312958843830733205051568, −1.02799315211862341221752386818, −0.810174228328889964159929270673, −0.19377922480347467975089559929,
0.19377922480347467975089559929, 0.810174228328889964159929270673, 1.02799315211862341221752386818, 1.07482312958843830733205051568, 1.71969344331057866791216969655, 1.76752898517719469551970079677, 1.83236468482907489337941901580, 2.28694975553724960250194204790, 2.68774532603071882282965502138, 2.69301788139933848461388439477, 2.91068982443971425012796922367, 3.12785674896288581037385834987, 3.21290243326620881805651351412, 3.23729871471545297064946763680, 3.80420662922011872959813061205, 3.88906274554816288996551382748, 4.06358808103318717592658499857, 4.18567303099730619734674541181, 4.41425959121698848034570837368, 4.43331353546425010104741425265, 4.58526595759162207004122506029, 4.83810840398826483059235908956, 4.89024839856780463284394815877, 5.16595645460576386564824575078, 5.34000673346758254706633579584
Plot not available for L-functions of degree greater than 10.
