| L(s) = 1 | + (−0.212 + 0.977i)2-s + (2.30 − 1.72i)3-s + (−0.909 − 0.415i)4-s + (0.975 + 2.01i)5-s + (1.19 + 2.61i)6-s + (−0.218 + 3.05i)7-s + (0.599 − 0.800i)8-s + (1.48 − 5.07i)9-s + (−2.17 + 0.525i)10-s + (−0.582 + 0.906i)11-s + (−2.81 + 0.611i)12-s + (−1.36 + 0.0976i)13-s + (−2.93 − 0.863i)14-s + (5.71 + 2.95i)15-s + (0.654 + 0.755i)16-s + (2.17 − 5.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.690i)2-s + (1.33 − 0.995i)3-s + (−0.454 − 0.207i)4-s + (0.436 + 0.899i)5-s + (0.488 + 1.06i)6-s + (−0.0826 + 1.15i)7-s + (0.211 − 0.283i)8-s + (0.496 − 1.69i)9-s + (−0.687 + 0.166i)10-s + (−0.175 + 0.273i)11-s + (−0.811 + 0.176i)12-s + (−0.378 + 0.0270i)13-s + (−0.785 − 0.230i)14-s + (1.47 + 0.762i)15-s + (0.163 + 0.188i)16-s + (0.526 − 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60725 + 0.455760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60725 + 0.455760i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.212 - 0.977i)T \) |
| 5 | \( 1 + (-0.975 - 2.01i)T \) |
| 23 | \( 1 + (3.89 + 2.80i)T \) |
| good | 3 | \( 1 + (-2.30 + 1.72i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (0.218 - 3.05i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (0.582 - 0.906i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.36 - 0.0976i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-2.17 + 5.82i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-2.12 + 4.66i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (6.11 - 2.79i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.518 + 3.60i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-4.08 - 7.48i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (10.3 - 3.03i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.88 + 2.52i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-1.06 + 1.06i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.45 - 0.318i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (4.89 + 4.23i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (7.99 - 1.15i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-13.3 - 2.89i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-2.34 + 1.50i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-12.8 + 4.79i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (0.149 - 0.172i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (6.43 - 3.51i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (1.55 - 10.8i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (5.50 + 3.00i)T + (52.4 + 81.6i)T^{2} \) |
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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
−12.55162711484801610421549016366, −11.53973767793406586574562664913, −9.786288733290883891803474738020, −9.258167312715467408887456607780, −8.183185539656630841332090840255, −7.29737514620197616045529541886, −6.58856853865138850558581778748, −5.28522873387356113920565839540, −3.09989411055627821206560441328, −2.21061209477496604939997490853,
1.79397802587328088344708277885, 3.56886595093912844132143561000, 4.13681525014329132836894964179, 5.53044392810782755893529039694, 7.74504386008125675808170477003, 8.356998986903736166173351227213, 9.473086826819951801309640200940, 10.01207401353463947056290868774, 10.71478961856925548363524192303, 12.24828394247202137410037043102
