| L(s) = 1 | + (0.5 − 1.32i)2-s + (−1.50 − 1.32i)4-s + 2.64i·7-s + (−2.50 + 1.32i)8-s + 3·9-s − 4·11-s + (3.50 + 1.32i)14-s + (0.500 + 3.96i)16-s + (1.5 − 3.96i)18-s + (−2 + 5.29i)22-s − 5.29i·23-s − 5·25-s + (3.50 − 3.96i)28-s − 10.5i·29-s + (5.50 + 1.32i)32-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.935i)2-s + (−0.750 − 0.661i)4-s + 0.999i·7-s + (−0.883 + 0.467i)8-s + 9-s − 1.20·11-s + (0.935 + 0.353i)14-s + (0.125 + 0.992i)16-s + (0.353 − 0.935i)18-s + (−0.426 + 1.12i)22-s − 1.10i·23-s − 25-s + (0.661 − 0.749i)28-s − 1.96i·29-s + (0.972 + 0.233i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.800900 - 0.482318i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.800900 - 0.482318i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.5 + 1.32i)T \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 5.29iT - 23T^{2} \) |
| 29 | \( 1 + 10.5iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.29iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 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
−15.12951767717978981967122694928, −13.64269436863226217674293583777, −12.75138234501504294993937161120, −11.83304773630415367156406193917, −10.49478530213177647399054753952, −9.541016767140040973811360193759, −8.078991164281878014507133708164, −5.94378865687961195754566585876, −4.50547011561856594800494603632, −2.47526344280229400760333690349,
3.86980794953892404706401381919, 5.32424572040100304631311636353, 7.06618838234201662451282681267, 7.81834738779393037299948699003, 9.520018906037646861781861458269, 10.75161213019772195520814858253, 12.60658723932896391012029797303, 13.35037996981770931698615231154, 14.36190767075539612348683886818, 15.70342312379845539044273957666
