| L(s) = 1 | + 5.50·5-s − 10.3·7-s + 3.20·11-s − 22.0i·13-s + 3.70·17-s + (10.3 + 15.9i)19-s − 44.5·23-s + 5.34·25-s + 39.6i·29-s − 36.9i·31-s − 56.9·35-s − 24.6i·37-s + 65.1i·41-s − 27.0·43-s + 67.5·47-s + ⋯ |
| L(s) = 1 | + 1.10·5-s − 1.47·7-s + 0.291·11-s − 1.69i·13-s + 0.218·17-s + (0.544 + 0.838i)19-s − 1.93·23-s + 0.213·25-s + 1.36i·29-s − 1.19i·31-s − 1.62·35-s − 0.666i·37-s + 1.58i·41-s − 0.628·43-s + 1.43·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7319466871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7319466871\) |
| \(L(2)\) |
|
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 | \( 1 + (-10.3 - 15.9i)T \) |
| good | 5 | \( 1 - 5.50T + 25T^{2} \) |
| 7 | \( 1 + 10.3T + 49T^{2} \) |
| 11 | \( 1 - 3.20T + 121T^{2} \) |
| 13 | \( 1 + 22.0iT - 169T^{2} \) |
| 17 | \( 1 - 3.70T + 289T^{2} \) |
| 23 | \( 1 + 44.5T + 529T^{2} \) |
| 29 | \( 1 - 39.6iT - 841T^{2} \) |
| 31 | \( 1 + 36.9iT - 961T^{2} \) |
| 37 | \( 1 + 24.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 65.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 27.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 67.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 14.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 70.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 63.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 78.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 87.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 36.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 2.54iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 16.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 70.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 135. iT - 9.40e3T^{2} \) |
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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
−9.146381969971966637135004451449, −8.080285943278947159358570820702, −7.44094376674330743621379148102, −6.29080373626125136379139676754, −5.94795458732200838797115376055, −5.36925764664073377549938191673, −3.94696227869105428513164750311, −3.20683883919424569942477249801, −2.37031463938960186382914539562, −1.13030548491751447875323438943,
0.16690566166630706881135536339, 1.70357114778343027637272836266, 2.46154651510400093526928500963, 3.55602868247639340440668695189, 4.34065845533418323159416499860, 5.48338791067785020061217199509, 6.27463933632632378546963874030, 6.60124254464854730931820628872, 7.44222120337156624044922505577, 8.676731627914875117023567446778
