L(s) = 1 | + (−0.5 − 0.866i)5-s + (−1 + 1.73i)11-s + 3·17-s − 19-s + (−1.5 − 2.59i)23-s + (−0.499 + 0.866i)25-s + (−2 + 3.46i)29-s + (2.5 + 4.33i)31-s + 10·37-s + (−3 − 5.19i)41-s + (3 − 5.19i)43-s + (−4 + 6.92i)47-s + (3.5 + 6.06i)49-s + 3·53-s + 1.99·55-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (−0.301 + 0.522i)11-s + 0.727·17-s − 0.229·19-s + (−0.312 − 0.541i)23-s + (−0.0999 + 0.173i)25-s + (−0.371 + 0.643i)29-s + (0.449 + 0.777i)31-s + 1.64·37-s + (−0.468 − 0.811i)41-s + (0.457 − 0.792i)43-s + (−0.583 + 1.01i)47-s + (0.5 + 0.866i)49-s + 0.412·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.615692919\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.615692919\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.699157407487801783117826202595, −7.895713173417943716693383897678, −7.35312361971692527748573688796, −6.43557297735975774945411042782, −5.61619642531035342431065107657, −4.82008112447506116388268978524, −4.10288208409951118958109915722, −3.10984295732141777961756380764, −2.09177287399667536600513909418, −0.885367002401265269664088565825,
0.65920071246210068966079514178, 2.07715463233027668175503528762, 3.04917901351017251881082674340, 3.82710629090034865094885928173, 4.73146679433286267611047752682, 5.73163655044939700886347902003, 6.25002722702370821511111696866, 7.21259472537554617289025941703, 7.949927499266847783744645699148, 8.375644653711369405055821968656