| L(s) = 1 | + 4-s − 2·5-s − 2·7-s + 8·11-s − 3·16-s − 10·17-s − 10·19-s − 2·20-s − 2·25-s − 2·28-s − 4·31-s + 4·35-s + 4·41-s − 2·43-s + 8·44-s + 8·47-s − 6·49-s − 6·53-s − 16·55-s − 8·59-s − 7·64-s − 6·67-s − 10·68-s + 16·71-s − 4·73-s − 10·76-s − 16·77-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s − 0.755·7-s + 2.41·11-s − 3/4·16-s − 2.42·17-s − 2.29·19-s − 0.447·20-s − 2/5·25-s − 0.377·28-s − 0.718·31-s + 0.676·35-s + 0.624·41-s − 0.304·43-s + 1.20·44-s + 1.16·47-s − 6/7·49-s − 0.824·53-s − 2.15·55-s − 1.04·59-s − 7/8·64-s − 0.733·67-s − 1.21·68-s + 1.89·71-s − 0.468·73-s − 1.14·76-s − 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22667121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22667121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 23 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 174 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.236894940328681490349463751696, −7.47999663485229153950745150913, −7.45628207362026958811955238762, −6.86661376067862079835013714337, −6.60563809391477060886885176333, −6.37431779766072415823128053425, −6.20321255043038000091092093698, −5.87486803901909635103121246176, −4.88241401673267097157112147684, −4.63286602880339918002755242818, −4.21973882520259127965942231713, −4.07798704613464740384160847535, −3.63718877556621655915960971546, −3.26626789349989049246207610809, −2.51820408223425551393347195049, −2.00296136672965993878470295531, −1.96848977286954044755835615890, −1.09009137398915744377303008917, 0, 0,
1.09009137398915744377303008917, 1.96848977286954044755835615890, 2.00296136672965993878470295531, 2.51820408223425551393347195049, 3.26626789349989049246207610809, 3.63718877556621655915960971546, 4.07798704613464740384160847535, 4.21973882520259127965942231713, 4.63286602880339918002755242818, 4.88241401673267097157112147684, 5.87486803901909635103121246176, 6.20321255043038000091092093698, 6.37431779766072415823128053425, 6.60563809391477060886885176333, 6.86661376067862079835013714337, 7.45628207362026958811955238762, 7.47999663485229153950745150913, 8.236894940328681490349463751696
