L(s) = 1 | + (0.5 + 0.240i)2-s + (0.178 − 0.0859i)3-s + (−1.05 − 1.32i)4-s + (0.445 + 1.94i)5-s + 0.109·6-s − 2.55·7-s + (−0.455 − 1.99i)8-s + (−1.84 + 2.31i)9-s + (−0.246 + 1.08i)10-s + (2.95 − 3.70i)11-s + (−0.301 − 0.145i)12-s + (−0.143 − 0.626i)13-s + (−1.27 − 0.615i)14-s + (0.246 + 0.309i)15-s + (−0.500 + 2.19i)16-s + (−0.246 + 1.08i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.170i)2-s + (0.103 − 0.0496i)3-s + (−0.527 − 0.661i)4-s + (0.199 + 0.872i)5-s + 0.0448·6-s − 0.965·7-s + (−0.161 − 0.706i)8-s + (−0.615 + 0.771i)9-s + (−0.0781 + 0.342i)10-s + (0.891 − 1.11i)11-s + (−0.0871 − 0.0419i)12-s + (−0.0396 − 0.173i)13-s + (−0.341 − 0.164i)14-s + (0.0637 + 0.0799i)15-s + (−0.125 + 0.547i)16-s + (−0.0599 + 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.836284 + 0.0330587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.836284 + 0.0330587i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (5.45 + 3.63i)T \) |
good | 2 | \( 1 + (-0.5 - 0.240i)T + (1.24 + 1.56i)T^{2} \) |
| 3 | \( 1 + (-0.178 + 0.0859i)T + (1.87 - 2.34i)T^{2} \) |
| 5 | \( 1 + (-0.445 - 1.94i)T + (-4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + (-2.95 + 3.70i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.143 + 0.626i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.246 - 1.08i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 + (-3.33 - 4.18i)T + (-4.22 + 18.5i)T^{2} \) |
| 23 | \( 1 + (-3.27 + 4.10i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (-0.821 - 0.395i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (5.37 + 2.58i)T + (19.3 + 24.2i)T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + (-6.99 - 3.36i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (0.623 + 0.781i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (1.72 - 7.54i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-1.85 + 8.12i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-7.11 + 3.42i)T + (38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + (-0.291 - 0.364i)T + (-14.9 + 65.3i)T^{2} \) |
| 71 | \( 1 + (-5.91 - 7.41i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-2.82 - 12.3i)T + (-65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + 5.09T + 79T^{2} \) |
| 83 | \( 1 + (13.4 - 6.47i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-15.3 + 7.38i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 12.9i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−15.93672698548587131640098840640, −14.44275034862139380622590988641, −14.06510352917959927194420407864, −12.84351815882347094861860630719, −11.08358939721922990004017426802, −10.03487273941397967342955116996, −8.709266190012255589339118589365, −6.68470187212503501191963663146, −5.62629719788149797121490619666, −3.40179336612502443232279256047,
3.45672301403236084943702316106, 5.06587383033595705286424851821, 7.00385334769351368487781017863, 9.002748540916390794843037881396, 9.396344263823800360130217675790, 11.73766216422731739604462584160, 12.54912843581988426019599765062, 13.45183587899407524681287726772, 14.68167199433446833216270308425, 16.10148772697550860910506054052