| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 2·7-s − 4·8-s − 3·9-s − 4·10-s − 6·12-s − 2·13-s + 4·14-s − 4·15-s + 5·16-s + 8·17-s + 6·18-s + 2·19-s + 6·20-s + 4·21-s − 2·23-s + 8·24-s + 3·25-s + 4·26-s + 14·27-s − 6·28-s + 8·30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s − 9-s − 1.26·10-s − 1.73·12-s − 0.554·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s + 1.94·17-s + 1.41·18-s + 0.458·19-s + 1.34·20-s + 0.872·21-s − 0.417·23-s + 1.63·24-s + 3/5·25-s + 0.784·26-s + 2.69·27-s − 1.13·28-s + 1.46·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T - 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22 T + 267 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 28 T + 362 T^{2} + 28 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.50901428243533981388364086178, −7.40822335760396283526976106681, −6.76480522608684036184089124470, −6.70607673978891963069265732972, −6.11073024375040691970898673995, −6.10241713976649208912922375505, −5.55988060780869967644727257781, −5.44092763489852699412642181937, −5.02381906710377050033545206810, −4.85299517677108427706779932115, −3.80327169940936606400986485839, −3.50238681027965812610083168346, −3.18585464916773625430629668266, −2.78995075912620420923943364598, −2.23489239307226375874739187761, −1.97848347504732522514644293968, −1.10804353101182654681588083505, −0.994718781229834995507983955976, 0, 0,
0.994718781229834995507983955976, 1.10804353101182654681588083505, 1.97848347504732522514644293968, 2.23489239307226375874739187761, 2.78995075912620420923943364598, 3.18585464916773625430629668266, 3.50238681027965812610083168346, 3.80327169940936606400986485839, 4.85299517677108427706779932115, 5.02381906710377050033545206810, 5.44092763489852699412642181937, 5.55988060780869967644727257781, 6.10241713976649208912922375505, 6.11073024375040691970898673995, 6.70607673978891963069265732972, 6.76480522608684036184089124470, 7.40822335760396283526976106681, 7.50901428243533981388364086178
