| L(s) = 1 | + 0.618·3-s − 1.61·7-s − 2.61·9-s + 5.52·11-s + 5.52·13-s − 3.41·17-s − 3.41·19-s − 1.00·21-s + 2.38·23-s − 3.47·27-s + 1.09·29-s − 5.52·31-s + 3.41·33-s + 8.93·37-s + 3.41·39-s + 5.85·41-s + 12.3·43-s − 1.09·47-s − 4.38·49-s − 2.10·51-s − 11.0·53-s − 2.10·57-s + 12.3·59-s + 1.14·61-s + 4.23·63-s + 10.4·67-s + 1.47·69-s + ⋯ |
| L(s) = 1 | + 0.356·3-s − 0.611·7-s − 0.872·9-s + 1.66·11-s + 1.53·13-s − 0.827·17-s − 0.782·19-s − 0.218·21-s + 0.496·23-s − 0.668·27-s + 0.202·29-s − 0.991·31-s + 0.593·33-s + 1.46·37-s + 0.546·39-s + 0.914·41-s + 1.87·43-s − 0.159·47-s − 0.625·49-s − 0.295·51-s − 1.51·53-s − 0.279·57-s + 1.60·59-s + 0.146·61-s + 0.533·63-s + 1.27·67-s + 0.177·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.102395043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.102395043\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 13 | \( 1 - 5.52T + 13T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + 5.52T + 31T^{2} \) |
| 37 | \( 1 - 8.93T + 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 8.93T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.690754167628291179070583703941, −7.83445030736295760183264390230, −6.79385440978038162047591959003, −6.24102307809836736127562372181, −5.79249713901146991253597860079, −4.39183998420279698967653721832, −3.82675483175360087059479578380, −3.06061574201469835412014948086, −2.01478706557906705398709420007, −0.829507448402520658462341994900,
0.829507448402520658462341994900, 2.01478706557906705398709420007, 3.06061574201469835412014948086, 3.82675483175360087059479578380, 4.39183998420279698967653721832, 5.79249713901146991253597860079, 6.24102307809836736127562372181, 6.79385440978038162047591959003, 7.83445030736295760183264390230, 8.690754167628291179070583703941
