| L(s) = 1 | − 2·3-s + 2·9-s − 2·13-s − 4·17-s − 2·19-s − 4·23-s − 6·27-s + 4·29-s + 12·31-s + 14·37-s + 4·39-s + 16·41-s − 8·43-s − 16·47-s + 6·49-s + 8·51-s + 18·53-s + 4·57-s + 12·59-s + 2·67-s + 8·69-s + 12·71-s + 4·73-s + 4·79-s + 11·81-s + 12·83-s − 8·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2/3·9-s − 0.554·13-s − 0.970·17-s − 0.458·19-s − 0.834·23-s − 1.15·27-s + 0.742·29-s + 2.15·31-s + 2.30·37-s + 0.640·39-s + 2.49·41-s − 1.21·43-s − 2.33·47-s + 6/7·49-s + 1.12·51-s + 2.47·53-s + 0.529·57-s + 1.56·59-s + 0.244·67-s + 0.963·69-s + 1.42·71-s + 0.468·73-s + 0.450·79-s + 11/9·81-s + 1.31·83-s − 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.525240762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525240762\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.000318243927244412241493002194, −7.951956100464347261885647380209, −7.10789445256684449422882857212, −7.01534723583488512453311132202, −6.51590854524446689831588539645, −6.35074982519372952450521305361, −6.00688224291904777763459890752, −5.71612796365569555938854972902, −5.13182787424182788764781439161, −5.02033174922278548344752523825, −4.50813257554485122453799114476, −4.12798405490171664212855804463, −4.02580455789628371515384611215, −3.43892332154625690865626492180, −2.61672370085233845960822954835, −2.45091634558485059536688940810, −2.22937810963164411499677669882, −1.35692292105796677658428698719, −0.805933288473836091323714604893, −0.43734075643188688902378163419,
0.43734075643188688902378163419, 0.805933288473836091323714604893, 1.35692292105796677658428698719, 2.22937810963164411499677669882, 2.45091634558485059536688940810, 2.61672370085233845960822954835, 3.43892332154625690865626492180, 4.02580455789628371515384611215, 4.12798405490171664212855804463, 4.50813257554485122453799114476, 5.02033174922278548344752523825, 5.13182787424182788764781439161, 5.71612796365569555938854972902, 6.00688224291904777763459890752, 6.35074982519372952450521305361, 6.51590854524446689831588539645, 7.01534723583488512453311132202, 7.10789445256684449422882857212, 7.951956100464347261885647380209, 8.000318243927244412241493002194
