| L(s) = 1 | + 8·16-s − 8·31-s + 72·41-s − 72·71-s + 34·81-s − 72·101-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 2·16-s − 1.43·31-s + 11.2·41-s − 8.54·71-s + 34/9·81-s − 7.16·101-s − 2.90·121-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 + 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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.083767216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.083767216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| good | 2 | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | \( ( 1 - 17 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 34 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 263 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 1897 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 9 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 2017 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 5257 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 8722 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 9 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 457 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 12097 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 148 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 18334 T^{4} + p^{4} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.33671372160898456626081487508, −4.26722962749976135008584376533, −4.18267941779906246362184575577, −4.11771674675733190737888020596, −4.03450727455848983956399887030, −3.89500042815600717433245623148, −3.70746263876546697099738404867, −3.64631049003216661322872059117, −3.19231420601490891700382420957, −3.00338921324611947263027893202, −2.98871528193082281661117938780, −2.97054957013135475697376262720, −2.83330462122790778181866696575, −2.70679070640550724305651403556, −2.43379318552621592066320106472, −2.33754420579424186017576341972, −2.17260563962186581615995065607, −1.84340842523270487923656087311, −1.83870234786733357720675780586, −1.26261431381247641569756370987, −1.21849906516767806716604929369, −1.18848292494883336989450164649, −1.05955377742613912421608884310, −0.57904563875377422959986203556, −0.32213802240742126134234867514,
0.32213802240742126134234867514, 0.57904563875377422959986203556, 1.05955377742613912421608884310, 1.18848292494883336989450164649, 1.21849906516767806716604929369, 1.26261431381247641569756370987, 1.83870234786733357720675780586, 1.84340842523270487923656087311, 2.17260563962186581615995065607, 2.33754420579424186017576341972, 2.43379318552621592066320106472, 2.70679070640550724305651403556, 2.83330462122790778181866696575, 2.97054957013135475697376262720, 2.98871528193082281661117938780, 3.00338921324611947263027893202, 3.19231420601490891700382420957, 3.64631049003216661322872059117, 3.70746263876546697099738404867, 3.89500042815600717433245623148, 4.03450727455848983956399887030, 4.11771674675733190737888020596, 4.18267941779906246362184575577, 4.26722962749976135008584376533, 4.33671372160898456626081487508
Plot not available for L-functions of degree greater than 10.
