L(s) = 1 | + 1.41i·2-s + (−0.495 − 2.95i)3-s − 2.00·4-s + 8.96i·5-s + (4.18 − 0.701i)6-s − 9.25·7-s − 2.82i·8-s + (−8.50 + 2.93i)9-s − 12.6·10-s − 17.1i·11-s + (0.991 + 5.91i)12-s + 21.0·13-s − 13.0i·14-s + (26.5 − 4.44i)15-s + 4.00·16-s − 15.0i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.165 − 0.986i)3-s − 0.500·4-s + 1.79i·5-s + (0.697 − 0.116i)6-s − 1.32·7-s − 0.353i·8-s + (−0.945 + 0.325i)9-s − 1.26·10-s − 1.56i·11-s + (0.0826 + 0.493i)12-s + 1.61·13-s − 0.934i·14-s + (1.76 − 0.296i)15-s + 0.250·16-s − 0.886i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.500836 - 0.423911i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.500836 - 0.423911i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (0.495 + 2.95i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 5 | \( 1 - 8.96iT - 25T^{2} \) |
| 7 | \( 1 + 9.25T + 49T^{2} \) |
| 11 | \( 1 + 17.1iT - 121T^{2} \) |
| 13 | \( 1 - 21.0T + 169T^{2} \) |
| 17 | \( 1 + 15.0iT - 289T^{2} \) |
| 19 | \( 1 + 11.0T + 361T^{2} \) |
| 23 | \( 1 + 38.9iT - 529T^{2} \) |
| 29 | \( 1 + 8.28iT - 841T^{2} \) |
| 31 | \( 1 - 31.3T + 961T^{2} \) |
| 37 | \( 1 + 30.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 27.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 61.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 14.5iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 5.82T + 3.72e3T^{2} \) |
| 67 | \( 1 + 73.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 46.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 85.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 13.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 32.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 32.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 88.8T + 9.40e3T^{2} \) |
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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
−10.95535533058716492536817077540, −10.35255545271985579768533716325, −8.870405185221542331942860873906, −8.075653741154927801009604353937, −6.70712965771954565875536030037, −6.53527327273505567064148822049, −5.85861727358275072243093079550, −3.52384703881647592514543202191, −2.80094935154180547213937534173, −0.31623201792488229004870149551,
1.48015798966483168248125232937, 3.52675815968927942734303142446, 4.27378490772324755379342149569, 5.26790470735227465965449363678, 6.30734263987615198268739795304, 8.237732028065450244640766590881, 8.998277263623941778405043194927, 9.676865506695905961022677792628, 10.27895316863561942360157439284, 11.53820816579711583549299472316