| L(s) = 1 | + 2.58·3-s + 204.·7-s − 236.·9-s + 271.·11-s + 952.·13-s + 1.17e3·17-s + 2.12e3·19-s + 529.·21-s + 4.78e3·23-s − 1.24e3·27-s − 7.03e3·29-s − 7.02e3·31-s + 704.·33-s − 1.17e4·37-s + 2.46e3·39-s − 5.07e3·41-s + 1.83e4·43-s + 2.69e4·47-s + 2.50e4·49-s + 3.04e3·51-s − 1.01e4·53-s + 5.50e3·57-s + 1.28e4·59-s + 1.31e4·61-s − 4.83e4·63-s − 3.62e4·67-s + 1.23e4·69-s + ⋯ |
| L(s) = 1 | + 0.166·3-s + 1.57·7-s − 0.972·9-s + 0.677·11-s + 1.56·13-s + 0.985·17-s + 1.35·19-s + 0.262·21-s + 1.88·23-s − 0.327·27-s − 1.55·29-s − 1.31·31-s + 0.112·33-s − 1.40·37-s + 0.259·39-s − 0.471·41-s + 1.51·43-s + 1.77·47-s + 1.48·49-s + 0.163·51-s − 0.496·53-s + 0.224·57-s + 0.479·59-s + 0.452·61-s − 1.53·63-s − 0.986·67-s + 0.313·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.626695186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.626695186\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.58T + 243T^{2} \) |
| 7 | \( 1 - 204.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 271.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 952.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.17e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.17e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.07e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.83e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.01e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.28e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.31e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.03e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.64e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.042807810318066094612322350200, −8.897692166358384110362894631170, −7.81271473273831895016685862292, −7.15403347073684187215685465395, −5.62595088258446042140697055608, −5.37694274800070822164240750653, −3.94613274878580903323400690402, −3.13764205827514863678881824758, −1.65369967671260392808621016884, −0.967001407707220006151917844482,
0.967001407707220006151917844482, 1.65369967671260392808621016884, 3.13764205827514863678881824758, 3.94613274878580903323400690402, 5.37694274800070822164240750653, 5.62595088258446042140697055608, 7.15403347073684187215685465395, 7.81271473273831895016685862292, 8.897692166358384110362894631170, 9.042807810318066094612322350200
