| L(s) = 1 | + 3·3-s − 1.12·5-s + 0.876·7-s + 9·9-s − 11·11-s + 71.0·13-s − 3.36·15-s + 22.2·17-s − 125.·19-s + 2.63·21-s − 104.·23-s − 123.·25-s + 27·27-s + 303.·29-s − 196.·31-s − 33·33-s − 0.984·35-s − 247.·37-s + 213.·39-s − 448.·41-s + 196.·43-s − 10.1·45-s + 28.5·47-s − 342.·49-s + 66.6·51-s + 38.4·53-s + 12.3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.100·5-s + 0.0473·7-s + 0.333·9-s − 0.301·11-s + 1.51·13-s − 0.0579·15-s + 0.316·17-s − 1.51·19-s + 0.0273·21-s − 0.948·23-s − 0.989·25-s + 0.192·27-s + 1.94·29-s − 1.14·31-s − 0.174·33-s − 0.00475·35-s − 1.10·37-s + 0.875·39-s − 1.70·41-s + 0.696·43-s − 0.0334·45-s + 0.0885·47-s − 0.997·49-s + 0.182·51-s + 0.0996·53-s + 0.0302·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 1.12T + 125T^{2} \) |
| 7 | \( 1 - 0.876T + 343T^{2} \) |
| 13 | \( 1 - 71.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 303.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 247.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 448.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 196.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 28.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 38.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 14.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 625.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 668.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 179.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 458.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 626.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 827.T + 9.12e5T^{2} \) |
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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
−8.411767325742697806933280181462, −7.82974569617000142788824232261, −6.70489132155268312508189300355, −6.14130110728928198635599652796, −5.10823079192003126314106126998, −4.03199578112362212484664935289, −3.49822120349542492535499791999, −2.30335863360138121773310135049, −1.42170646097473393085516616965, 0,
1.42170646097473393085516616965, 2.30335863360138121773310135049, 3.49822120349542492535499791999, 4.03199578112362212484664935289, 5.10823079192003126314106126998, 6.14130110728928198635599652796, 6.70489132155268312508189300355, 7.82974569617000142788824232261, 8.411767325742697806933280181462
