| L(s) = 1 | + i·2-s − 4-s + 2.01·5-s + (1.78 + 1.95i)7-s − i·8-s + 2.01i·10-s − i·11-s + (−1.95 + 1.78i)14-s + 16-s + 5.92·17-s − 0.657i·19-s − 2.01·20-s + 22-s + 7.87i·23-s − 0.935·25-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.901·5-s + (0.674 + 0.738i)7-s − 0.353i·8-s + 0.637i·10-s − 0.301i·11-s + (−0.522 + 0.476i)14-s + 0.250·16-s + 1.43·17-s − 0.150i·19-s − 0.450·20-s + 0.213·22-s + 1.64i·23-s − 0.187·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.086018432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086018432\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.78 - 1.95i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.01T + 5T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 19 | \( 1 + 0.657iT - 19T^{2} \) |
| 23 | \( 1 - 7.87iT - 23T^{2} \) |
| 29 | \( 1 - 1.32iT - 29T^{2} \) |
| 31 | \( 1 + 10.6iT - 31T^{2} \) |
| 37 | \( 1 - 5.87T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 + 1.56T + 43T^{2} \) |
| 47 | \( 1 - 0.532T + 47T^{2} \) |
| 53 | \( 1 - 7.44iT - 53T^{2} \) |
| 59 | \( 1 - 4.03T + 59T^{2} \) |
| 61 | \( 1 + 0.250iT - 61T^{2} \) |
| 67 | \( 1 + 7.81T + 67T^{2} \) |
| 71 | \( 1 + 0.609iT - 71T^{2} \) |
| 73 | \( 1 + 0.532iT - 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 5.22T + 89T^{2} \) |
| 97 | \( 1 - 2.71iT - 97T^{2} \) |
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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
−9.490287634749236902580162175001, −9.074150689739232067140494342814, −7.86403665839867295833346695349, −7.60478467019353465985745996021, −6.12528931148703414221772533010, −5.75966292830878564142809831419, −5.06134140911464057333975809389, −3.83616725137395097998160707075, −2.56112098894823565802724792774, −1.33006595297265707933620384409,
1.01976969820193861913893217472, 2.01019586586484744710829241741, 3.13976176847647145786769911347, 4.27062555326385726278166215689, 5.06435621358543080731623232745, 5.95479301861980291989321626238, 6.99473303409172527394417651159, 7.940112172960588469658862979738, 8.670626935592757783750502832031, 9.702228353093015400932007284894
