L(s) = 1 | − 8·43-s − 2·81-s + 8·103-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 + 233-s + 239-s + 241-s + 251-s + ⋯ |
L(s) = 1 | − 8·43-s − 2·81-s + 8·103-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 + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8087434306\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8087434306\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T^{8} \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 + T^{8} \) |
| 13 | \( ( 1 + T^{4} )^{2} \) |
good | 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 + T^{8} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( ( 1 + T^{2} )^{8} \) |
| 23 | \( ( 1 + T^{4} )^{4} \) |
| 29 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 31 | \( ( 1 + T^{2} )^{8} \) |
| 37 | \( ( 1 + T^{4} )^{4} \) |
| 41 | \( ( 1 + T^{8} )^{2} \) |
| 43 | \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \) |
| 47 | \( ( 1 + T^{8} )^{2} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{4} )^{4} \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( ( 1 + T^{8} )^{2} \) |
| 73 | \( ( 1 + T^{4} )^{4} \) |
| 79 | \( ( 1 + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{4} )^{4} \) |
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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
−4.25516670060208125221873554307, −4.14203036302488975250800946887, −3.88045409738657295570573515846, −3.87742855825956738592218482077, −3.69885892486212747686099568515, −3.35543705022065715252031079188, −3.35002569589319896260021749475, −3.33201940402259172931230067895, −3.26046804360944876249085346011, −3.25363107092413021471558226961, −3.13590189295346853960459546997, −2.93842527631062362337481380029, −2.72010490759924103216621579801, −2.36763761140940274544485370535, −2.31629495240296061589942592590, −2.28423962520395933815014765202, −2.06247559250676054708604681768, −1.80732844300405348404991807916, −1.80554722541465229422251812685, −1.78325175309551369784117433868, −1.35983823549704205282968685797, −1.32573228339451575568794014122, −1.21099946057303335924408527341, −0.65481148062080092977819628280, −0.50116809066366794890095799437,
0.50116809066366794890095799437, 0.65481148062080092977819628280, 1.21099946057303335924408527341, 1.32573228339451575568794014122, 1.35983823549704205282968685797, 1.78325175309551369784117433868, 1.80554722541465229422251812685, 1.80732844300405348404991807916, 2.06247559250676054708604681768, 2.28423962520395933815014765202, 2.31629495240296061589942592590, 2.36763761140940274544485370535, 2.72010490759924103216621579801, 2.93842527631062362337481380029, 3.13590189295346853960459546997, 3.25363107092413021471558226961, 3.26046804360944876249085346011, 3.33201940402259172931230067895, 3.35002569589319896260021749475, 3.35543705022065715252031079188, 3.69885892486212747686099568515, 3.87742855825956738592218482077, 3.88045409738657295570573515846, 4.14203036302488975250800946887, 4.25516670060208125221873554307
Plot not available for L-functions of degree greater than 10.