| L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s − 4·11-s − 2·13-s + 4·15-s + 8·17-s − 2·19-s − 4·21-s − 4·23-s − 4·25-s + 2·27-s − 12·31-s + 8·33-s − 4·35-s − 16·37-s + 4·39-s − 12·43-s − 4·47-s + 3·49-s − 16·51-s − 12·53-s + 8·55-s + 4·57-s − 2·59-s − 2·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s − 1.20·11-s − 0.554·13-s + 1.03·15-s + 1.94·17-s − 0.458·19-s − 0.872·21-s − 0.834·23-s − 4/5·25-s + 0.384·27-s − 2.15·31-s + 1.39·33-s − 0.676·35-s − 2.63·37-s + 0.640·39-s − 1.82·43-s − 0.583·47-s + 3/7·49-s − 2.24·51-s − 1.64·53-s + 1.07·55-s + 0.529·57-s − 0.260·59-s − 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T - 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 244 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 198 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.10981537603080613659292966041, −9.669178550281557358504751419493, −8.943232394600392933338263795138, −8.623966591954662412499846625440, −7.989047184897726615632956030175, −7.76778911373645919925843376403, −7.50275708361587723303325108619, −7.07205985075139445796822359770, −6.23500742017146899702803340536, −5.97597297706520474597493477799, −5.28461326205475223784605876009, −5.22495632374253093676830123624, −4.83013321738864696019833486483, −4.04389016047341696325705537080, −3.39408808875610188540011035469, −3.21570982092036343526160577808, −2.01625353173241807079879469062, −1.61290835970329585108104720285, 0, 0,
1.61290835970329585108104720285, 2.01625353173241807079879469062, 3.21570982092036343526160577808, 3.39408808875610188540011035469, 4.04389016047341696325705537080, 4.83013321738864696019833486483, 5.22495632374253093676830123624, 5.28461326205475223784605876009, 5.97597297706520474597493477799, 6.23500742017146899702803340536, 7.07205985075139445796822359770, 7.50275708361587723303325108619, 7.76778911373645919925843376403, 7.989047184897726615632956030175, 8.623966591954662412499846625440, 8.943232394600392933338263795138, 9.669178550281557358504751419493, 10.10981537603080613659292966041
