| L(s) = 1 | + 0.162i·5-s + 0.418·7-s + 5.50i·11-s − 2.39i·17-s − 4.35·19-s + 3.33i·23-s + 4.97·25-s + 0.0679i·35-s − 11.8·43-s + 11.4i·47-s − 6.82·49-s − 0.893·55-s − 10.8·61-s + 5.82·73-s + 2.30i·77-s + ⋯ |
| L(s) = 1 | + 0.0725i·5-s + 0.158·7-s + 1.65i·11-s − 0.581i·17-s − 1.00·19-s + 0.695i·23-s + 0.994·25-s + 0.0114i·35-s − 1.80·43-s + 1.66i·47-s − 0.974·49-s − 0.120·55-s − 1.38·61-s + 0.681·73-s + 0.262i·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.078732928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078732928\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35T \) |
| good | 5 | \( 1 - 0.162iT - 5T^{2} \) |
| 7 | \( 1 - 0.418T + 7T^{2} \) |
| 11 | \( 1 - 5.50iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 2.39iT - 17T^{2} \) |
| 23 | \( 1 - 3.33iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 11.4iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 5.14iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.203178742270806106536204763298, −8.281048797360071756248352277436, −7.51172991875383000545510535016, −6.87926613059486396986013741669, −6.14864592439399230978002765079, −4.90698593056038961973515209474, −4.62466774972607092688310263596, −3.44674017738261844730865851085, −2.39331544031075826617341648637, −1.46689295672020673130002087480,
0.34154935130658166619040734392, 1.66447033400597866886376969994, 2.89011427827274298587508320605, 3.66771990002340083600956870530, 4.64725275807389413665337674414, 5.49940627686557536398747962301, 6.32423421952426976776560950533, 6.85673475518821081568050369582, 8.168101943104172133996065159723, 8.411237767603831571815721234304
