| L(s) = 1 | − 2·2-s + 4·3-s + 2·4-s − 8·6-s − 2·7-s − 5·8-s + 13·9-s + 3·11-s + 8·12-s + 9·13-s + 4·14-s + 5·16-s − 7·17-s − 26·18-s + 5·19-s − 8·21-s − 6·22-s + 9·23-s − 20·24-s − 18·26-s + 30·27-s − 4·28-s − 2·31-s + 2·32-s + 12·33-s + 14·34-s + 26·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 4-s − 3.26·6-s − 0.755·7-s − 1.76·8-s + 13/3·9-s + 0.904·11-s + 2.30·12-s + 2.49·13-s + 1.06·14-s + 5/4·16-s − 1.69·17-s − 6.12·18-s + 1.14·19-s − 1.74·21-s − 1.27·22-s + 1.87·23-s − 4.08·24-s − 3.53·26-s + 5.77·27-s − 0.755·28-s − 0.359·31-s + 0.353·32-s + 2.08·33-s + 2.40·34-s + 13/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.909590826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.909590826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2:C_4$ | \( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 - 4 T + p T^{2} + 10 T^{3} - 29 T^{4} + 10 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2^2:C_4$ | \( 1 - 9 T + 48 T^{2} - 235 T^{3} + 1011 T^{4} - 235 p T^{5} + 48 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 7 T + 52 T^{2} + 245 T^{3} + 1311 T^{4} + 245 p T^{5} + 52 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 5 T + 21 T^{2} - 145 T^{3} + 956 T^{4} - 145 p T^{5} + 21 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 9 T + 13 T^{2} + 15 T^{3} + 196 T^{4} + 15 p T^{5} + 13 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 11 T^{2} + 90 T^{3} + 661 T^{4} + 90 p T^{5} + 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 2 T - 27 T^{2} - 116 T^{3} + 605 T^{4} - 116 p T^{5} - 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 3 T - 28 T^{2} + 195 T^{3} + 451 T^{4} + 195 p T^{5} - 28 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 8 T + 23 T^{2} - 356 T^{3} + 3905 T^{4} - 356 p T^{5} + 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 13 T + 67 T^{2} + 115 T^{3} - 3864 T^{4} + 115 p T^{5} + 67 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 14 T + 23 T^{2} + 400 T^{3} - 2899 T^{4} + 400 p T^{5} + 23 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 15 T + 206 T^{2} + 1965 T^{3} + 18601 T^{4} + 1965 p T^{5} + 206 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 17 T + 42 T^{2} - 1025 T^{3} - 12559 T^{4} - 1025 p T^{5} + 42 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 13 T + 133 T^{2} - 1541 T^{3} + 17940 T^{4} - 1541 p T^{5} + 133 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 11 T - 27 T^{2} - 515 T^{3} - 364 T^{4} - 515 p T^{5} - 27 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 15 T + 56 T^{2} - 675 T^{3} + 11821 T^{4} - 675 p T^{5} + 56 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 24 T + 173 T^{2} - 360 T^{3} + 361 T^{4} - 360 p T^{5} + 173 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 79 T^{2} - 420 T^{3} + 7501 T^{4} - 420 p T^{5} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 12 T - 3 T^{2} - 1220 T^{3} - 13419 T^{4} - 1220 p T^{5} - 3 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.60619893015399852595274898410, −7.58916840781567480713542472030, −7.39796928524933603185869173064, −6.78790601892638465569724300265, −6.74182960071738236698096780789, −6.69214340012714718286081077116, −6.60128414191754673246401795741, −6.00616421777245108332414444289, −5.89465061465269953966573264347, −5.80461334124547507520502094582, −5.04913067712906438222701195764, −4.85638837827987416973398228099, −4.46525016061961142568519730494, −4.38050307894760852969614669969, −3.88457692234076674833481806039, −3.65215652827796834547697465072, −3.47209440438846098574587591040, −3.20209536969952486444914093886, −3.06701549372289046179734573894, −2.55781075699805860835860405198, −2.21145176730470897871207667766, −1.98792515149183940013774022582, −1.47600992649402937578513303485, −1.04098764733397903278137786464, −0.74835790526217055645186908335,
0.74835790526217055645186908335, 1.04098764733397903278137786464, 1.47600992649402937578513303485, 1.98792515149183940013774022582, 2.21145176730470897871207667766, 2.55781075699805860835860405198, 3.06701549372289046179734573894, 3.20209536969952486444914093886, 3.47209440438846098574587591040, 3.65215652827796834547697465072, 3.88457692234076674833481806039, 4.38050307894760852969614669969, 4.46525016061961142568519730494, 4.85638837827987416973398228099, 5.04913067712906438222701195764, 5.80461334124547507520502094582, 5.89465061465269953966573264347, 6.00616421777245108332414444289, 6.60128414191754673246401795741, 6.69214340012714718286081077116, 6.74182960071738236698096780789, 6.78790601892638465569724300265, 7.39796928524933603185869173064, 7.58916840781567480713542472030, 7.60619893015399852595274898410
