| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 4·8-s + 4·10-s + 2·11-s − 8·13-s + 5·16-s − 10·17-s − 4·19-s + 6·20-s + 4·22-s + 2·23-s − 5·25-s − 16·26-s + 4·31-s + 6·32-s − 20·34-s + 8·37-s − 8·38-s + 8·40-s − 2·41-s + 6·44-s + 4·46-s − 10·47-s − 10·50-s − 24·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s + 0.603·11-s − 2.21·13-s + 5/4·16-s − 2.42·17-s − 0.917·19-s + 1.34·20-s + 0.852·22-s + 0.417·23-s − 25-s − 3.13·26-s + 0.718·31-s + 1.06·32-s − 3.42·34-s + 1.31·37-s − 1.29·38-s + 1.26·40-s − 0.312·41-s + 0.904·44-s + 0.589·46-s − 1.45·47-s − 1.41·50-s − 3.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 101 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 129 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 207 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 26 T + 355 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.32999472949357098542493411408, −6.93914546361762739765234555412, −6.66783281430768160092629832549, −6.48248190684088067162223539998, −6.03380219116260970618320059631, −6.02072372443107454468346866447, −5.23177298879784910581648850926, −5.04873968736802034067633114709, −4.76325978676030346993989445654, −4.46621911429606557408742049254, −4.00648231924344030377544372363, −3.94628670646552474249604495882, −3.11034096175556012318085943646, −2.72589083661860230681705531101, −2.39925681400888753539627544740, −2.29112392372240368614616774625, −1.58788065413354132868664297900, −1.43008990398391180455392037500, 0, 0,
1.43008990398391180455392037500, 1.58788065413354132868664297900, 2.29112392372240368614616774625, 2.39925681400888753539627544740, 2.72589083661860230681705531101, 3.11034096175556012318085943646, 3.94628670646552474249604495882, 4.00648231924344030377544372363, 4.46621911429606557408742049254, 4.76325978676030346993989445654, 5.04873968736802034067633114709, 5.23177298879784910581648850926, 6.02072372443107454468346866447, 6.03380219116260970618320059631, 6.48248190684088067162223539998, 6.66783281430768160092629832549, 6.93914546361762739765234555412, 7.32999472949357098542493411408
