| L(s) = 1 | + 2-s + 1.63·3-s + 4-s + 1.19·5-s + 1.63·6-s + 2.04·7-s + 8-s − 0.314·9-s + 1.19·10-s + 3.68·11-s + 1.63·12-s + 2.15·13-s + 2.04·14-s + 1.95·15-s + 16-s − 6.53·17-s − 0.314·18-s + 5.31·19-s + 1.19·20-s + 3.34·21-s + 3.68·22-s − 5.89·23-s + 1.63·24-s − 3.57·25-s + 2.15·26-s − 5.43·27-s + 2.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.946·3-s + 0.5·4-s + 0.533·5-s + 0.669·6-s + 0.772·7-s + 0.353·8-s − 0.104·9-s + 0.377·10-s + 1.11·11-s + 0.473·12-s + 0.598·13-s + 0.546·14-s + 0.505·15-s + 0.250·16-s − 1.58·17-s − 0.0740·18-s + 1.21·19-s + 0.266·20-s + 0.730·21-s + 0.785·22-s − 1.22·23-s + 0.334·24-s − 0.714·25-s + 0.422·26-s − 1.04·27-s + 0.386·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.434359265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.434359265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.63T + 3T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 - 5.31T + 19T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 31 | \( 1 - 8.99T + 31T^{2} \) |
| 37 | \( 1 - 1.82T + 37T^{2} \) |
| 41 | \( 1 + 8.32T + 41T^{2} \) |
| 43 | \( 1 + 3.68T + 43T^{2} \) |
| 47 | \( 1 + 0.992T + 47T^{2} \) |
| 53 | \( 1 + 5.66T + 53T^{2} \) |
| 59 | \( 1 - 2.94T + 59T^{2} \) |
| 61 | \( 1 + 1.80T + 61T^{2} \) |
| 67 | \( 1 - 6.04T + 67T^{2} \) |
| 71 | \( 1 - 3.75T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 2.05T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 4.59T + 97T^{2} \) |
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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
−9.359040869595843417007408847630, −8.386537829090893130308126217018, −8.008129838795058980962927027010, −6.75276013366007744042211150315, −6.17728403930610069001836203784, −5.16270059410328930653642064712, −4.21484132927120323830006674282, −3.45684747893461358866589462679, −2.34035037842746460097980257705, −1.56025530297488037727488772479,
1.56025530297488037727488772479, 2.34035037842746460097980257705, 3.45684747893461358866589462679, 4.21484132927120323830006674282, 5.16270059410328930653642064712, 6.17728403930610069001836203784, 6.75276013366007744042211150315, 8.008129838795058980962927027010, 8.386537829090893130308126217018, 9.359040869595843417007408847630
