| L(s) = 1 | − 2·3-s − 6·7-s + 2·9-s + 12·21-s + 2·23-s − 6·27-s + 24·41-s + 18·43-s + 14·47-s + 18·49-s + 16·61-s − 12·63-s + 6·67-s − 4·69-s + 11·81-s − 22·83-s + 36·101-s − 18·103-s + 26·107-s + 22·121-s − 48·123-s + 127-s − 36·129-s + 131-s + 137-s + 139-s − 28·141-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.26·7-s + 2/3·9-s + 2.61·21-s + 0.417·23-s − 1.15·27-s + 3.74·41-s + 2.74·43-s + 2.04·47-s + 18/7·49-s + 2.04·61-s − 1.51·63-s + 0.733·67-s − 0.481·69-s + 11/9·81-s − 2.41·83-s + 3.58·101-s − 1.77·103-s + 2.51·107-s + 2·121-s − 4.32·123-s + 0.0887·127-s − 3.16·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.35·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079918778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079918778\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
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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
−9.604994132106875945435616786468, −9.378468980958517742437818312667, −8.908489870436310052046291335253, −8.676683686322070143775019249456, −7.67013848081013162138557875546, −7.58391356396562637697442744786, −7.09655167360036780894579923693, −6.79580121814763522725822710827, −6.18119996349557034845715173650, −5.92255289785252634124047291822, −5.73648311835224548715439573185, −5.38943781518919203046266981553, −4.35635188458020468337971909209, −4.29392024536477669311303045219, −3.74131552264969916001740768214, −3.17512173195372653193629663407, −2.56100977990067252553594394880, −2.24603293672649670424589853110, −0.78258933533769630609459500687, −0.66370069951966065907765649761,
0.66370069951966065907765649761, 0.78258933533769630609459500687, 2.24603293672649670424589853110, 2.56100977990067252553594394880, 3.17512173195372653193629663407, 3.74131552264969916001740768214, 4.29392024536477669311303045219, 4.35635188458020468337971909209, 5.38943781518919203046266981553, 5.73648311835224548715439573185, 5.92255289785252634124047291822, 6.18119996349557034845715173650, 6.79580121814763522725822710827, 7.09655167360036780894579923693, 7.58391356396562637697442744786, 7.67013848081013162138557875546, 8.676683686322070143775019249456, 8.908489870436310052046291335253, 9.378468980958517742437818312667, 9.604994132106875945435616786468
