| L(s) = 1 | − 4·9-s + 20·25-s − 32·29-s − 4·49-s + 10·81-s + 32·89-s − 64·101-s + 80·109-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 4·25-s − 5.94·29-s − 4/7·49-s + 10/9·81-s + 3.39·89-s − 6.36·101-s + 7.66·109-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2055916884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2055916884\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 11 | \( ( 1 + 4 T^{2} - 74 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 12 T^{2} + 54 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 12 T^{2} + 294 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 36 T^{2} + 726 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 84 T^{2} + 3366 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 68 T^{2} + 3574 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 60 T^{2} + 1718 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 140 T^{2} + 8998 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 12 T^{2} + 5334 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 76 T^{2} + 3286 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 100 T^{2} + 8598 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 84 T^{2} + 3846 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 92 T^{2} + 4774 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T^{2} + 7366 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 140 T^{2} + 13558 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 8 T + 174 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 348 T^{2} + 48774 T^{4} - 348 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.59961487055788513142883227930, −3.53851328469592283140656663003, −3.19705419827112640910412111105, −3.13347089716237170088303211738, −3.13041805752683372960276622141, −3.09235170974157437649652225183, −3.06432455753452887780437356378, −2.64383193191801504327738617005, −2.50744089403050685802798627024, −2.47956796927206798772765813454, −2.41516096600347117903864632739, −2.36764686123470836899735458760, −2.08625739674617074945546291251, −1.90518862900206837407888683667, −1.83193433561513129362219935652, −1.73964228743306611186992977777, −1.72814982967549599472302998927, −1.29869974982056437147758812830, −1.23627735633287766125558613042, −1.17708515573153106192031468404, −0.889806557150383880791720508645, −0.69933200024096527359283663420, −0.61469656229331487006964328508, −0.23166405886334208986077462208, −0.05628399986262734661042056584,
0.05628399986262734661042056584, 0.23166405886334208986077462208, 0.61469656229331487006964328508, 0.69933200024096527359283663420, 0.889806557150383880791720508645, 1.17708515573153106192031468404, 1.23627735633287766125558613042, 1.29869974982056437147758812830, 1.72814982967549599472302998927, 1.73964228743306611186992977777, 1.83193433561513129362219935652, 1.90518862900206837407888683667, 2.08625739674617074945546291251, 2.36764686123470836899735458760, 2.41516096600347117903864632739, 2.47956796927206798772765813454, 2.50744089403050685802798627024, 2.64383193191801504327738617005, 3.06432455753452887780437356378, 3.09235170974157437649652225183, 3.13041805752683372960276622141, 3.13347089716237170088303211738, 3.19705419827112640910412111105, 3.53851328469592283140656663003, 3.59961487055788513142883227930
Plot not available for L-functions of degree greater than 10.
