| L(s) = 1 | − 3-s + 1.27·5-s + 1.05·7-s + 9-s + 2.86·11-s − 0.442·13-s − 1.27·15-s + 7.11·17-s + 5.55·19-s − 1.05·21-s + 23-s − 3.37·25-s − 27-s + 29-s + 9.86·31-s − 2.86·33-s + 1.35·35-s + 3.13·37-s + 0.442·39-s + 0.570·41-s + 3.31·43-s + 1.27·45-s + 7.62·47-s − 5.87·49-s − 7.11·51-s − 0.0892·53-s + 3.65·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.570·5-s + 0.400·7-s + 0.333·9-s + 0.865·11-s − 0.122·13-s − 0.329·15-s + 1.72·17-s + 1.27·19-s − 0.231·21-s + 0.208·23-s − 0.674·25-s − 0.192·27-s + 0.185·29-s + 1.77·31-s − 0.499·33-s + 0.228·35-s + 0.515·37-s + 0.0708·39-s + 0.0890·41-s + 0.505·43-s + 0.190·45-s + 1.11·47-s − 0.839·49-s − 0.996·51-s − 0.0122·53-s + 0.493·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.572235990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.572235990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 0.442T + 13T^{2} \) |
| 17 | \( 1 - 7.11T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 31 | \( 1 - 9.86T + 31T^{2} \) |
| 37 | \( 1 - 3.13T + 37T^{2} \) |
| 41 | \( 1 - 0.570T + 41T^{2} \) |
| 43 | \( 1 - 3.31T + 43T^{2} \) |
| 47 | \( 1 - 7.62T + 47T^{2} \) |
| 53 | \( 1 + 0.0892T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 - 0.605T + 61T^{2} \) |
| 67 | \( 1 + 2.08T + 67T^{2} \) |
| 71 | \( 1 + 9.62T + 71T^{2} \) |
| 73 | \( 1 - 9.39T + 73T^{2} \) |
| 79 | \( 1 + 4.40T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 1.77T + 97T^{2} \) |
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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
−7.66575417343926053304881192488, −7.23912021445213070125538196466, −6.18577195409111424422141536746, −5.88405520912854984203443763590, −5.10757132427326860000740058318, −4.43598518350197037237944794318, −3.51444220128092206614781670333, −2.69322278893921265196700949623, −1.47389304756662621446749955226, −0.937894510332341865043757388756,
0.937894510332341865043757388756, 1.47389304756662621446749955226, 2.69322278893921265196700949623, 3.51444220128092206614781670333, 4.43598518350197037237944794318, 5.10757132427326860000740058318, 5.88405520912854984203443763590, 6.18577195409111424422141536746, 7.23912021445213070125538196466, 7.66575417343926053304881192488
