| L(s) = 1 | + 4-s + 2·7-s − 3·9-s + 12·19-s + 7·25-s + 2·28-s + 12·31-s − 3·36-s − 8·37-s + 16·43-s − 11·49-s + 48·61-s − 6·63-s − 64-s − 10·67-s + 42·73-s + 12·76-s − 16·79-s + 7·100-s + 18·103-s + 4·109-s − 22·121-s + 12·124-s + 127-s + 131-s + 24·133-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s − 9-s + 2.75·19-s + 7/5·25-s + 0.377·28-s + 2.15·31-s − 1/2·36-s − 1.31·37-s + 2.43·43-s − 1.57·49-s + 6.14·61-s − 0.755·63-s − 1/8·64-s − 1.22·67-s + 4.91·73-s + 1.37·76-s − 1.80·79-s + 7/10·100-s + 1.77·103-s + 0.383·109-s − 2·121-s + 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 2.08·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.928546844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.928546844\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 67 T^{2} + 2280 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 10 T^{2} - 3381 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 83 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 21 T + 220 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 130 T^{2} + 8979 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.04532853718794092714058461838, −7.03154918378976038834858519640, −6.80496670658953220724280009125, −6.51459236105846820909429063171, −6.41268935276135310844838135633, −5.84061067738401603450943865328, −5.83777947186473117199178573516, −5.67902050924829705499226361025, −5.13468123834843549994647832446, −5.09215214477851443991482740471, −5.01504153658695392158069780174, −4.92031240975501060690406989073, −4.18216248992123602678516571441, −4.17198804798731653661667790111, −3.90304057365746255171389376716, −3.32627435883242887293893509564, −3.27196116718546034265893205859, −3.09772901026072418877457341082, −2.66663255895280966722409705775, −2.48057068822045702189526436727, −2.02678563660850663516439690604, −1.93806784612429135672285063979, −1.12599623472628539730121699840, −0.849761486407505170143592379115, −0.78488720321465058855482677581,
0.78488720321465058855482677581, 0.849761486407505170143592379115, 1.12599623472628539730121699840, 1.93806784612429135672285063979, 2.02678563660850663516439690604, 2.48057068822045702189526436727, 2.66663255895280966722409705775, 3.09772901026072418877457341082, 3.27196116718546034265893205859, 3.32627435883242887293893509564, 3.90304057365746255171389376716, 4.17198804798731653661667790111, 4.18216248992123602678516571441, 4.92031240975501060690406989073, 5.01504153658695392158069780174, 5.09215214477851443991482740471, 5.13468123834843549994647832446, 5.67902050924829705499226361025, 5.83777947186473117199178573516, 5.84061067738401603450943865328, 6.41268935276135310844838135633, 6.51459236105846820909429063171, 6.80496670658953220724280009125, 7.03154918378976038834858519640, 7.04532853718794092714058461838
