| L(s) = 1 | − 2·9-s − 24·17-s + 16·25-s + 8·41-s + 8·49-s − 32·73-s + 3·81-s − 8·89-s + 8·97-s − 40·113-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 5.82·17-s + 16/5·25-s + 1.24·41-s + 8/7·49-s − 3.74·73-s + 1/3·81-s − 0.847·89-s + 0.812·97-s − 3.76·113-s − 2.54·121-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 + 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3740965821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3740965821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.89075077343429651326380444043, −6.53831527705011687095527797070, −6.44892099048159614654016620824, −6.08690711260584154577645668082, −5.86420289870004039399181292456, −5.85231550382732688502712607515, −5.41139737298482303099850265332, −5.05328345852048839220373226192, −5.01488211565020311430219594598, −4.59954440103965887802302073423, −4.56658796897671523681622390349, −4.28781478118713908139072943339, −4.14372872299324192402002588248, −4.09541131555695810006852240627, −3.56263225918027442201960691961, −3.05481220348225492357865504944, −3.03358164120905729934056503887, −2.69203294738754593302184304092, −2.46387766530499081724185067589, −2.31393935281418529212285917885, −2.03547712999414382622382547343, −1.60181474126501741136921801089, −1.22283087260145487921329597734, −0.68965853604184405565575827159, −0.14421751405532148046622919484,
0.14421751405532148046622919484, 0.68965853604184405565575827159, 1.22283087260145487921329597734, 1.60181474126501741136921801089, 2.03547712999414382622382547343, 2.31393935281418529212285917885, 2.46387766530499081724185067589, 2.69203294738754593302184304092, 3.03358164120905729934056503887, 3.05481220348225492357865504944, 3.56263225918027442201960691961, 4.09541131555695810006852240627, 4.14372872299324192402002588248, 4.28781478118713908139072943339, 4.56658796897671523681622390349, 4.59954440103965887802302073423, 5.01488211565020311430219594598, 5.05328345852048839220373226192, 5.41139737298482303099850265332, 5.85231550382732688502712607515, 5.86420289870004039399181292456, 6.08690711260584154577645668082, 6.44892099048159614654016620824, 6.53831527705011687095527797070, 6.89075077343429651326380444043
