L(s) = 1 | + 2.13i·2-s + (0.780 + 1.54i)3-s − 2.56·4-s + 3.09·5-s + (−3.30 + 1.66i)6-s − 1.19i·8-s + (−1.78 + 2.41i)9-s + 6.60i·10-s + (3.09 − 1.19i)11-s + (−1.99 − 3.96i)12-s + (3.30 − 1.44i)13-s + (2.41 + 4.78i)15-s − 2.56·16-s − 5.73·17-s + (−5.15 − 3.80i)18-s − 2.89·19-s + ⋯ |
L(s) = 1 | + 1.51i·2-s + (0.450 + 0.892i)3-s − 1.28·4-s + 1.38·5-s + (−1.34 + 0.680i)6-s − 0.424i·8-s + (−0.593 + 0.804i)9-s + 2.08i·10-s + (0.932 − 0.361i)11-s + (−0.577 − 1.14i)12-s + (0.915 − 0.401i)13-s + (0.623 + 1.23i)15-s − 0.640·16-s − 1.39·17-s + (−1.21 − 0.896i)18-s − 0.664·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.305260 + 1.89188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.305260 + 1.89188i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.780 - 1.54i)T \) |
| 11 | \( 1 + (-3.09 + 1.19i)T \) |
| 13 | \( 1 + (-3.30 + 1.44i)T \) |
good | 2 | \( 1 - 2.13iT - 2T^{2} \) |
| 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 + 3.09iT - 23T^{2} \) |
| 29 | \( 1 - 5.73T + 29T^{2} \) |
| 31 | \( 1 + 2.68iT - 31T^{2} \) |
| 37 | \( 1 + 9.56iT - 37T^{2} \) |
| 41 | \( 1 - 5.20iT - 41T^{2} \) |
| 43 | \( 1 + 6.60iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6.18iT - 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 2.89iT - 61T^{2} \) |
| 67 | \( 1 - 2.68iT - 67T^{2} \) |
| 71 | \( 1 - 4.82T + 71T^{2} \) |
| 73 | \( 1 - 9.49T + 73T^{2} \) |
| 79 | \( 1 - 0.813iT - 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 6.56T + 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−11.19680582181002234086645886828, −10.47630789973116085676561950396, −9.260786890792501828769140251877, −8.915960176333266902721900294749, −8.061625464346358714562381379581, −6.51128813524498410207647402092, −6.14102541593701549923311433394, −5.07685108022023931374497165290, −4.05143327431403482923805097792, −2.31915819369036140944735008121,
1.42741217595853875768635085012, 2.04876498815253760925841427505, 3.23690412595832400616877492588, 4.54349408108782796442090277006, 6.30087731450453804094258030292, 6.69599288987505727735322998739, 8.512963063336923890770108613775, 9.168310990690605206383882264050, 9.845162641519136233735462029608, 10.90304382838359176277507114503