| L(s) = 1 | + (−2.5 + 0.866i)7-s + 1.73i·13-s + 8.66i·19-s − 5·25-s + 10.3i·31-s + 37-s − 8·43-s + (5.5 − 4.33i)49-s + 8.66i·61-s + 11·67-s − 1.73i·73-s − 13·79-s + (−1.49 − 4.33i)91-s − 19.0i·97-s + 19.0i·103-s + ⋯ |
| L(s) = 1 | + (−0.944 + 0.327i)7-s + 0.480i·13-s + 1.98i·19-s − 25-s + 1.86i·31-s + 0.164·37-s − 1.21·43-s + (0.785 − 0.618i)49-s + 1.10i·61-s + 1.34·67-s − 0.202i·73-s − 1.46·79-s + (−0.157 − 0.453i)91-s − 1.93i·97-s + 1.87i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.522220 + 0.733568i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.522220 + 0.733568i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 8.66iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8.66iT - 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
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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
−10.32175231359791411584049196486, −9.893086223222024123647772416995, −8.913751522216197291076461934527, −8.110573864603345765777075064195, −7.03622273857576276276741062693, −6.20130449939886660425278286750, −5.39362291706783301781115890897, −4.03543179677870772489110320726, −3.16107985935435432608510306972, −1.72953084860175627500825651548,
0.44567927555889145076557034814, 2.41460040215232219778713294450, 3.48618687265904621824860114667, 4.56272641846991815272137540972, 5.71078796891535215951667332307, 6.61656830984141797103630660508, 7.40031489141267915017225163522, 8.382134965287589616455486077094, 9.453109426050750789327109798882, 9.889813406078335955541434941801
