| L(s) = 1 | + 4-s − 6·7-s + 14·13-s + 18·19-s − 2·25-s − 6·28-s + 30·37-s + 20·43-s + 7·49-s + 14·52-s − 20·61-s − 64-s + 18·76-s + 40·79-s − 84·91-s − 24·97-s − 2·100-s − 4·103-s − 13·121-s + 127-s + 131-s − 108·133-s + 137-s + 139-s + 30·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.26·7-s + 3.88·13-s + 4.12·19-s − 2/5·25-s − 1.13·28-s + 4.93·37-s + 3.04·43-s + 49-s + 1.94·52-s − 2.56·61-s − 1/8·64-s + 2.06·76-s + 4.50·79-s − 8.80·91-s − 2.43·97-s − 1/5·100-s − 0.394·103-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s − 9.36·133-s + 0.0854·137-s + 0.0848·139-s + 2.46·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.792276680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.792276680\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 10 T^{2} - 741 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 15 T + 112 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 46 T^{2} + 435 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 26 T^{2} - 2805 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 47 T^{2} - 5712 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.06599627306686240145125445645, −6.57793391911260500854964573570, −6.48040843868636936986873011333, −6.33251606732403172073131024762, −6.08984291620356467315095532089, −5.93591200514890429025625800234, −5.66267584015963010698415677818, −5.65860325515406083485264027347, −5.58343557697314435769037803976, −4.83080780722921812066666495590, −4.65088387459145165618235996187, −4.55815013585362866585935570210, −4.01334308360745916121543265177, −3.75420410534121744530193952027, −3.71566828556201169461871162381, −3.46179666406888936734040917164, −3.21100186546361693844877729837, −2.91116792953719528751354218605, −2.81706533892856470132188510100, −2.49996177567096719999162076363, −2.07041439750048115798518459712, −1.31003228291903418009309784104, −1.13799281355123868310687181497, −1.04771875890462447784402793125, −0.62533679537498635433639099711,
0.62533679537498635433639099711, 1.04771875890462447784402793125, 1.13799281355123868310687181497, 1.31003228291903418009309784104, 2.07041439750048115798518459712, 2.49996177567096719999162076363, 2.81706533892856470132188510100, 2.91116792953719528751354218605, 3.21100186546361693844877729837, 3.46179666406888936734040917164, 3.71566828556201169461871162381, 3.75420410534121744530193952027, 4.01334308360745916121543265177, 4.55815013585362866585935570210, 4.65088387459145165618235996187, 4.83080780722921812066666495590, 5.58343557697314435769037803976, 5.65860325515406083485264027347, 5.66267584015963010698415677818, 5.93591200514890429025625800234, 6.08984291620356467315095532089, 6.33251606732403172073131024762, 6.48040843868636936986873011333, 6.57793391911260500854964573570, 7.06599627306686240145125445645
