L(s) = 1 | − 2·16-s − 40·31-s + 64·61-s − 20·121-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 + 233-s + 239-s + 241-s + ⋯ |
L(s) = 1 | − 1/2·16-s − 7.18·31-s + 8.19·61-s − 1.81·121-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 + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.270937470\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.270937470\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 5 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 337 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 47 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 311 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 5 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 217 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4249 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4174 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 8302 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 9214 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 157 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 13294 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 11422 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−3.94802401076731583449641794564, −3.79887924173657110817517578154, −3.75203418997097855914935939764, −3.71241398545032039715353974247, −3.67804464933517959546343540395, −3.52706953563382768068521759879, −3.39520106862618747619567531958, −3.30346549876554805154311316790, −3.11186551165872092388329979531, −2.88128404095386189726962634435, −2.59477042384432548923997772541, −2.58469100551412304650657732011, −2.44641920364435099892125134425, −2.40685977297862836793042327186, −2.05766479338680668056488943355, −2.05442375292574260023157002983, −1.77966917527211038749105900184, −1.69167693584895768397947699139, −1.66797192772868878072795962864, −1.60458891110859627493327031266, −1.08181168992992866993607110932, −0.805511050533989392355855800935, −0.67489116149060552623711061002, −0.51566491372296849110385127954, −0.18295184168711939073708203378,
0.18295184168711939073708203378, 0.51566491372296849110385127954, 0.67489116149060552623711061002, 0.805511050533989392355855800935, 1.08181168992992866993607110932, 1.60458891110859627493327031266, 1.66797192772868878072795962864, 1.69167693584895768397947699139, 1.77966917527211038749105900184, 2.05442375292574260023157002983, 2.05766479338680668056488943355, 2.40685977297862836793042327186, 2.44641920364435099892125134425, 2.58469100551412304650657732011, 2.59477042384432548923997772541, 2.88128404095386189726962634435, 3.11186551165872092388329979531, 3.30346549876554805154311316790, 3.39520106862618747619567531958, 3.52706953563382768068521759879, 3.67804464933517959546343540395, 3.71241398545032039715353974247, 3.75203418997097855914935939764, 3.79887924173657110817517578154, 3.94802401076731583449641794564
Plot not available for L-functions of degree greater than 10.