L(s) = 1 | + (0.398 − 0.230i)2-s + (−2.88 − 1.66i)3-s + (−0.894 + 1.54i)4-s + (1.49 + 1.66i)5-s − 1.53·6-s + (0.0944 + 0.0545i)7-s + 1.74i·8-s + (4.04 + 7.00i)9-s + (0.978 + 0.319i)10-s + 2.27·11-s + (5.15 − 2.97i)12-s + (2.22 + 1.28i)13-s + 0.0501·14-s + (−1.54 − 7.28i)15-s + (−1.38 − 2.40i)16-s + (−4.60 + 2.66i)17-s + ⋯ |
L(s) = 1 | + (0.281 − 0.162i)2-s + (−1.66 − 0.961i)3-s + (−0.447 + 0.774i)4-s + (0.668 + 0.743i)5-s − 0.625·6-s + (0.0356 + 0.0206i)7-s + 0.616i·8-s + (1.34 + 2.33i)9-s + (0.309 + 0.100i)10-s + 0.685·11-s + (1.48 − 0.859i)12-s + (0.616 + 0.356i)13-s + 0.0134·14-s + (−0.397 − 1.88i)15-s + (−0.346 − 0.600i)16-s + (−1.11 + 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.682705 + 0.333752i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.682705 + 0.333752i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.49 - 1.66i)T \) |
| 37 | \( 1 + (-1.93 - 5.76i)T \) |
good | 2 | \( 1 + (-0.398 + 0.230i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (2.88 + 1.66i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.0944 - 0.0545i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 + (-2.22 - 1.28i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (4.60 - 2.66i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.75 - 3.03i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 41 | \( 1 + (-5.48 + 9.49i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 4.53iT - 43T^{2} \) |
| 47 | \( 1 + 4.28iT - 47T^{2} \) |
| 53 | \( 1 + (-5.36 + 3.09i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.03 + 5.26i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.57 - 6.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.76 - 5.63i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.53 - 2.66i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.89iT - 73T^{2} \) |
| 79 | \( 1 + (5.44 - 9.43i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.36 + 1.36i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.51 + 11.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.82iT - 97T^{2} \) |
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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
−12.74396251700653373007638053209, −11.70967703889907095136233203148, −11.21064926785826610533122597538, −10.14737852544052062195334123575, −8.570557671293409367725770085954, −7.21067365471020511427853415306, −6.43205937239544007565331376980, −5.50459387771482354908149875265, −4.07199061505361958848995982395, −1.92327093638218010107767806328,
0.825732596980726088867814585893, 4.39422833381586418523478651084, 4.80191064192881252523339014696, 6.04810686350968319550991077863, 6.45100204706517535105538036533, 9.015758585490434810696025826527, 9.546287647515486052703016127533, 10.60885183328514118160806606081, 11.24580819567664964610701137844, 12.48118414829979931515908780755