| L(s) = 1 | + (−1.09 + 0.896i)2-s + (0.307 + 0.672i)3-s + (0.392 − 1.96i)4-s + (−2.03 − 2.35i)5-s + (−0.938 − 0.460i)6-s + (3.56 + 2.28i)7-s + (1.32 + 2.49i)8-s + (1.60 − 1.85i)9-s + (4.33 + 0.745i)10-s + (2.09 − 0.300i)11-s + (1.43 − 0.338i)12-s + (0.883 + 1.37i)13-s + (−5.94 + 0.689i)14-s + (0.956 − 2.09i)15-s + (−3.69 − 1.54i)16-s + (1.50 + 5.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.773 + 0.633i)2-s + (0.177 + 0.388i)3-s + (0.196 − 0.980i)4-s + (−0.911 − 1.05i)5-s + (−0.383 − 0.187i)6-s + (1.34 + 0.865i)7-s + (0.469 + 0.882i)8-s + (0.535 − 0.618i)9-s + (1.37 + 0.235i)10-s + (0.631 − 0.0907i)11-s + (0.415 − 0.0976i)12-s + (0.244 + 0.381i)13-s + (−1.58 + 0.184i)14-s + (0.246 − 0.540i)15-s + (−0.922 − 0.385i)16-s + (0.365 + 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.875950 + 0.229147i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.875950 + 0.229147i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.09 - 0.896i)T \) |
| 23 | \( 1 + (3.27 + 3.50i)T \) |
| good | 3 | \( 1 + (-0.307 - 0.672i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (2.03 + 2.35i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-3.56 - 2.28i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-2.09 + 0.300i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.883 - 1.37i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.50 - 5.13i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.23 + 4.22i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.283 - 0.964i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.95 - 1.80i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (1.33 - 1.53i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (8.34 + 9.63i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.55 - 0.708i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 3.79iT - 47T^{2} \) |
| 53 | \( 1 + (8.33 + 5.35i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (5.75 - 3.70i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.60 - 3.51i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (10.7 + 1.54i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-9.41 - 1.35i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.996 - 0.292i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-1.50 + 0.969i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-9.64 - 8.36i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (12.5 - 5.75i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (10.4 - 9.03i)T + (13.8 - 96.0i)T^{2} \) |
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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
−12.33026551977993785033262297547, −11.74410657282073116545758574928, −10.62709572210000240711190088259, −9.258029865471077889496506218482, −8.594605276378591482592517679020, −8.025175978901866172319432555281, −6.56023959447746169371461176627, −5.12082133299906221313026005042, −4.21439391315999697275359962635, −1.43692927192593042016567120100,
1.53901066825062823739100262889, 3.30774822797328733056034258562, 4.49679244769250315846603185175, 6.89591600215176934061653296309, 7.77075384239490963802066587443, 8.022571129742613499389949589929, 9.822919722798691311686385164798, 10.67743981202113011272802057692, 11.46876359997288533719290917689, 12.04683965075932395361263838286
