| L(s) = 1 | + 16-s + 48·41-s + 32·61-s + 96·71-s + 7·81-s − 24·101-s − 88·121-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 | + 1/4·16-s + 7.49·41-s + 4.09·61-s + 11.3·71-s + 7/9·81-s − 2.38·101-s − 8·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.250007256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.250007256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( 1 - 7 T^{4} - 32 T^{8} - 7 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + p T^{2} )^{8} \) |
| 13 | \( ( 1 - 337 T^{4} + p^{4} T^{8} )( 1 + 191 T^{4} + p^{4} T^{8} ) \) |
| 17 | \( 1 - 383 T^{4} + 63168 T^{8} - 383 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( 1 - 98 T^{4} - 270237 T^{8} - 98 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 46 T^{2} + 1275 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 + 626 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 12 T + 89 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 - 1778 T^{4} - 257517 T^{8} - 1778 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 2302 T^{4} + 419523 T^{8} + 2302 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 4078 T^{4} - 3521037 T^{8} + 4078 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 24 T + 263 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 5617 T^{4} + 3152448 T^{8} + 5617 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 12791 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 103 T^{2} + 2688 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 2207 T^{4} - 83658432 T^{8} - 2207 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−4.21529641687458827264341497819, −4.17854004506861383690138939692, −3.99357518365721313163860900712, −3.87006814168156895008408541210, −3.72696705082544811428387730906, −3.67314540833866296092125350837, −3.66716916458013185420928661777, −3.64441949432510936807831562513, −3.36613949444810844123080570748, −2.92890533691762644413039925698, −2.82488158900620323320571645911, −2.67970205679519785361093339440, −2.48483487106442689931828185303, −2.45123182907009929689869705329, −2.44275498090225032607503511862, −2.28744905374245201937040402328, −2.28521819819940436824847564018, −1.81642820284614951193297965701, −1.74080078547885216288148132063, −1.19198183968524442303711609186, −1.18288363846510395820940051922, −1.06679443780472622036014041243, −0.828801129102139930902204029920, −0.61791874041307553190663448562, −0.39854930195171868114534590144,
0.39854930195171868114534590144, 0.61791874041307553190663448562, 0.828801129102139930902204029920, 1.06679443780472622036014041243, 1.18288363846510395820940051922, 1.19198183968524442303711609186, 1.74080078547885216288148132063, 1.81642820284614951193297965701, 2.28521819819940436824847564018, 2.28744905374245201937040402328, 2.44275498090225032607503511862, 2.45123182907009929689869705329, 2.48483487106442689931828185303, 2.67970205679519785361093339440, 2.82488158900620323320571645911, 2.92890533691762644413039925698, 3.36613949444810844123080570748, 3.64441949432510936807831562513, 3.66716916458013185420928661777, 3.67314540833866296092125350837, 3.72696705082544811428387730906, 3.87006814168156895008408541210, 3.99357518365721313163860900712, 4.17854004506861383690138939692, 4.21529641687458827264341497819
Plot not available for L-functions of degree greater than 10.
