| L(s) = 1 | − i·2-s − 4-s − 1.18i·5-s + (1.74 − 1.99i)7-s + i·8-s − 1.18·10-s + (2.25 + 2.43i)11-s + 5.98·13-s + (−1.99 − 1.74i)14-s + 16-s − 4.29·17-s − 0.203·19-s + 1.18i·20-s + (2.43 − 2.25i)22-s + 3.11·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.530i·5-s + (0.658 − 0.752i)7-s + 0.353i·8-s − 0.374·10-s + (0.678 + 0.734i)11-s + 1.65·13-s + (−0.532 − 0.465i)14-s + 0.250·16-s − 1.04·17-s − 0.0467·19-s + 0.265i·20-s + (0.519 − 0.479i)22-s + 0.648·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0273 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0273 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.889746419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.889746419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.74 + 1.99i)T \) |
| 11 | \( 1 + (-2.25 - 2.43i)T \) |
| good | 5 | \( 1 + 1.18iT - 5T^{2} \) |
| 13 | \( 1 - 5.98T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 + 0.203T + 19T^{2} \) |
| 23 | \( 1 - 3.11T + 23T^{2} \) |
| 29 | \( 1 - 2.50iT - 29T^{2} \) |
| 31 | \( 1 + 2.79iT - 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 4.37iT - 43T^{2} \) |
| 47 | \( 1 + 2.05iT - 47T^{2} \) |
| 53 | \( 1 + 1.51T + 53T^{2} \) |
| 59 | \( 1 + 7.96iT - 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 2.29T + 73T^{2} \) |
| 79 | \( 1 + 9.85iT - 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 8.87iT - 89T^{2} \) |
| 97 | \( 1 + 9.46iT - 97T^{2} \) |
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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
−9.286890319551952714612226572745, −8.765593780517114579545485235730, −7.973634658396729112605712519000, −6.94242148457194137576995600227, −6.06151335356171904208025878865, −4.72784790853770094384768517332, −4.34103123900759996736573564842, −3.31575023333918235906164403739, −1.81111508210757684306317725641, −0.982074709887220821197337725130,
1.25851401376644170375423752184, 2.77864771808097253432586642884, 3.86058906652281199219407182293, 4.81214906307409595101348005211, 5.95268932108935541621355556434, 6.32288144383573177468991667938, 7.25348564003296930789212394518, 8.393023893905394153700227355496, 8.699511903226325376930714860207, 9.439857527215945488591261102074
