L(s) = 1 | + 4-s + 4·7-s + 3·9-s + 8·11-s + 25-s + 4·28-s + 3·36-s − 2·41-s + 8·44-s + 24·47-s + 18·49-s − 12·61-s + 12·63-s − 64-s + 16·67-s + 24·71-s − 24·73-s + 32·77-s − 12·79-s + 9·81-s + 24·83-s + 24·99-s + 100-s − 48·101-s − 12·109-s − 24·113-s + 20·121-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 1.51·7-s + 9-s + 2.41·11-s + 1/5·25-s + 0.755·28-s + 1/2·36-s − 0.312·41-s + 1.20·44-s + 3.50·47-s + 18/7·49-s − 1.53·61-s + 1.51·63-s − 1/8·64-s + 1.95·67-s + 2.84·71-s − 2.80·73-s + 3.64·77-s − 1.35·79-s + 81-s + 2.63·83-s + 2.41·99-s + 1/10·100-s − 4.77·101-s − 1.14·109-s − 2.25·113-s + 1.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.614427669\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.614427669\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1814 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 86 T^{2} + 3579 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 74 T^{2} + 4875 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 79 T^{2} + 3432 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 102 T^{2} + 6923 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 166 T^{2} + 1416 T^{3} + 13131 T^{4} + 1416 p T^{5} + 166 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 70 T^{2} - 832 T^{3} + 12955 T^{4} - 832 p T^{5} + 70 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 24 T + 278 T^{2} - 3168 T^{3} + 34107 T^{4} - 3168 p T^{5} + 278 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 + 114 T^{2} + 5075 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 284 T^{2} + 38214 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.074611455716876630249729725974, −8.071074847714670305018357280877, −7.80847589949252843462671563237, −7.46765208270832339609442075487, −7.29298679907247890969640831212, −6.89967061345871209793607213911, −6.82652890835511270867855548992, −6.57923784371057094640366166827, −6.44117077034275467537782467939, −5.85124107676889981100956137911, −5.67721242917518920918660514439, −5.58944656747658198271444944586, −5.04389541842113208352719404677, −4.78868192794662394033183142343, −4.60128968403003840717569991260, −4.03991580650060646356779476920, −4.03982223024861026039479772306, −3.69348136732323047037140457206, −3.67422605828793524993035547406, −2.66793085885086306255511574276, −2.54089270158444987710601766190, −2.23793044652720837979118737590, −1.50620310225230971187876536333, −1.31110380170542219094179035408, −1.07679203176650816575375594695,
1.07679203176650816575375594695, 1.31110380170542219094179035408, 1.50620310225230971187876536333, 2.23793044652720837979118737590, 2.54089270158444987710601766190, 2.66793085885086306255511574276, 3.67422605828793524993035547406, 3.69348136732323047037140457206, 4.03982223024861026039479772306, 4.03991580650060646356779476920, 4.60128968403003840717569991260, 4.78868192794662394033183142343, 5.04389541842113208352719404677, 5.58944656747658198271444944586, 5.67721242917518920918660514439, 5.85124107676889981100956137911, 6.44117077034275467537782467939, 6.57923784371057094640366166827, 6.82652890835511270867855548992, 6.89967061345871209793607213911, 7.29298679907247890969640831212, 7.46765208270832339609442075487, 7.80847589949252843462671563237, 8.071074847714670305018357280877, 8.074611455716876630249729725974