| L(s) = 1 | − 4·3-s + 10·9-s − 32·27-s + 12·41-s + 40·43-s + 28·49-s − 28·67-s + 89·81-s − 36·83-s + 36·89-s − 12·107-s + 14·121-s − 48·123-s + 127-s − 160·129-s + 131-s + 137-s + 139-s − 112·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 10/3·9-s − 6.15·27-s + 1.87·41-s + 6.09·43-s + 4·49-s − 3.42·67-s + 89/9·81-s − 3.95·83-s + 3.81·89-s − 1.16·107-s + 1.27·121-s − 4.32·123-s + 0.0887·127-s − 14.0·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 9.23·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610357683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610357683\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 142 T^{2} + 14835 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.50197198095000856121783184517, −6.34186270191794645930235771020, −6.07015332757248056055328264271, −5.97084569661213222854921567087, −5.88102577926111353205268279583, −5.78898511509479424494239835075, −5.47403497463139711757498571672, −5.36739073328481432068376946444, −5.05066732546330666237271840380, −4.77425300252919880412751809457, −4.36192491862872156380671793395, −4.22395190518720324456351522912, −4.15345079921294696344276114530, −4.11626779992499838603258934124, −3.66744709256206539299879682834, −3.50767862226550132018734544717, −2.85186641827609316091127985000, −2.54169160458573523186853681394, −2.53551104521227321984338282045, −2.25259962310594989455302925968, −1.59414790499809998539044405142, −1.53663954806448441323743265768, −0.936932629245459087448355313027, −0.66375882600806276559657655504, −0.45991367095403731573251505221,
0.45991367095403731573251505221, 0.66375882600806276559657655504, 0.936932629245459087448355313027, 1.53663954806448441323743265768, 1.59414790499809998539044405142, 2.25259962310594989455302925968, 2.53551104521227321984338282045, 2.54169160458573523186853681394, 2.85186641827609316091127985000, 3.50767862226550132018734544717, 3.66744709256206539299879682834, 4.11626779992499838603258934124, 4.15345079921294696344276114530, 4.22395190518720324456351522912, 4.36192491862872156380671793395, 4.77425300252919880412751809457, 5.05066732546330666237271840380, 5.36739073328481432068376946444, 5.47403497463139711757498571672, 5.78898511509479424494239835075, 5.88102577926111353205268279583, 5.97084569661213222854921567087, 6.07015332757248056055328264271, 6.34186270191794645930235771020, 6.50197198095000856121783184517
