| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s + 4·17-s − 8·23-s + 12·25-s + 2·27-s + 20·29-s + 36-s − 8·43-s − 10·49-s − 8·51-s − 24·53-s − 4·61-s − 64-s + 4·68-s + 16·69-s − 24·75-s − 4·81-s − 40·87-s − 8·92-s + 12·100-s − 4·101-s − 64·103-s − 16·107-s + 2·108-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s − 0.577·12-s + 0.970·17-s − 1.66·23-s + 12/5·25-s + 0.384·27-s + 3.71·29-s + 1/6·36-s − 1.21·43-s − 1.42·49-s − 1.12·51-s − 3.29·53-s − 0.512·61-s − 1/8·64-s + 0.485·68-s + 1.92·69-s − 2.77·75-s − 4/9·81-s − 4.28·87-s − 0.834·92-s + 6/5·100-s − 0.398·101-s − 6.30·103-s − 1.54·107-s + 0.192·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3518016054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3518016054\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 2 T^{2} - 357 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 + 102 T^{2} + 6923 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 142 T^{2} + 12243 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 50 T^{2} - 6909 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.89035366029564923123428802745, −6.81152903187538649792713412488, −6.53091056816621393154791539079, −6.44696323468050826457177793133, −6.38540297357788949139437156721, −6.29624559472414710370640791856, −5.59158091462670474381159881405, −5.51761029466553066029773442556, −5.32117634263532148084616147780, −5.24153708373649841930163576622, −4.82890356872882950882885569913, −4.59716699201408474500974769669, −4.42880489427841828127163397409, −4.23209384963344659878017277029, −3.89049741399223370027194030881, −3.42522957090486783690852490411, −3.17735851809504378545212035113, −2.85088454661961585487477555243, −2.81503036708817073237438680705, −2.58199019299838039445052739756, −1.98203221996889590288056288733, −1.38927280885199105712802525945, −1.33176286387248386211811266400, −1.08959055642862441260669015721, −0.15234640115352282176719972337,
0.15234640115352282176719972337, 1.08959055642862441260669015721, 1.33176286387248386211811266400, 1.38927280885199105712802525945, 1.98203221996889590288056288733, 2.58199019299838039445052739756, 2.81503036708817073237438680705, 2.85088454661961585487477555243, 3.17735851809504378545212035113, 3.42522957090486783690852490411, 3.89049741399223370027194030881, 4.23209384963344659878017277029, 4.42880489427841828127163397409, 4.59716699201408474500974769669, 4.82890356872882950882885569913, 5.24153708373649841930163576622, 5.32117634263532148084616147780, 5.51761029466553066029773442556, 5.59158091462670474381159881405, 6.29624559472414710370640791856, 6.38540297357788949139437156721, 6.44696323468050826457177793133, 6.53091056816621393154791539079, 6.81152903187538649792713412488, 6.89035366029564923123428802745
