L(s) = 1 | + (2.62 + 2.62i)2-s + (−3.11 − 3.11i)3-s + 9.78i·4-s − 4.53i·5-s − 16.3i·6-s − 0.745·7-s + (−15.1 + 15.1i)8-s + 10.3i·9-s + (11.8 − 11.8i)10-s + (0.0423 + 0.0423i)11-s + (30.4 − 30.4i)12-s + 8.30i·13-s + (−1.95 − 1.95i)14-s + (−14.0 + 14.0i)15-s − 40.5·16-s + (15.2 + 15.2i)17-s + ⋯ |
L(s) = 1 | + (1.31 + 1.31i)2-s + (−1.03 − 1.03i)3-s + 2.44i·4-s − 0.906i·5-s − 2.72i·6-s − 0.106·7-s + (−1.89 + 1.89i)8-s + 1.15i·9-s + (1.18 − 1.18i)10-s + (0.00384 + 0.00384i)11-s + (2.53 − 2.53i)12-s + 0.638i·13-s + (−0.139 − 0.139i)14-s + (−0.939 + 0.939i)15-s − 2.53·16-s + (0.897 + 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.17947 + 0.566144i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17947 + 0.566144i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-22.0 + 18.7i)T \) |
good | 2 | \( 1 + (-2.62 - 2.62i)T + 4iT^{2} \) |
| 3 | \( 1 + (3.11 + 3.11i)T + 9iT^{2} \) |
| 5 | \( 1 + 4.53iT - 25T^{2} \) |
| 7 | \( 1 + 0.745T + 49T^{2} \) |
| 11 | \( 1 + (-0.0423 - 0.0423i)T + 121iT^{2} \) |
| 13 | \( 1 - 8.30iT - 169T^{2} \) |
| 17 | \( 1 + (-15.2 - 15.2i)T + 289iT^{2} \) |
| 19 | \( 1 + (25.1 + 25.1i)T + 361iT^{2} \) |
| 23 | \( 1 - 8.64T + 529T^{2} \) |
| 31 | \( 1 + (5.09 + 5.09i)T + 961iT^{2} \) |
| 37 | \( 1 + (-10.0 + 10.0i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + (-2.50 + 2.50i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (11.2 + 11.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (23.0 - 23.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 42.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 106.T + 3.48e3T^{2} \) |
| 61 | \( 1 + (42.3 + 42.3i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 - 75.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 71.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (73.7 - 73.7i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + (78.7 + 78.7i)T + 6.24e3iT^{2} \) |
| 83 | \( 1 - 15.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + (22.8 + 22.8i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-42.8 + 42.8i)T - 9.40e3iT^{2} \) |
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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
−16.99952974484881863574139918421, −16.00049277620544984180669140048, −14.59168759056489398741254473071, −13.08327757038801006960131246071, −12.72551103291777718119671978392, −11.56175081813304305707853331337, −8.444407355063947988627836793240, −6.97323286213651169553456736135, −5.96827886354922175232360218838, −4.64144976502120482743189231835,
3.32172985133567089676984978740, 4.89520948468533016460411422647, 6.16609247298252896762696064335, 10.02815282008404774869610468227, 10.59178990238765604463886484887, 11.57957881352770016883215074148, 12.68292357972146377991168409744, 14.31782902983355190523849846866, 15.10938512165633292155529386112, 16.46933018907083941855576230510