L(s) = 1 | + 2.07i·2-s − 0.0823·3-s − 2.29·4-s − 3.08·5-s − 0.170i·6-s − 3.12i·7-s − 0.601i·8-s − 2.99·9-s − 6.39i·10-s − 2.26i·11-s + 0.188·12-s − 0.833i·13-s + 6.48·14-s + 0.254·15-s − 3.33·16-s + 3.85·17-s + ⋯ |
L(s) = 1 | + 1.46i·2-s − 0.0475·3-s − 1.14·4-s − 1.38·5-s − 0.0696i·6-s − 1.18i·7-s − 0.212i·8-s − 0.997·9-s − 2.02i·10-s − 0.683i·11-s + 0.0544·12-s − 0.231i·13-s + 1.73·14-s + 0.0656·15-s − 0.833·16-s + 0.935·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.159150 - 0.110305i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159150 - 0.110305i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 349 | \( 1 + (6.55 + 17.4i)T \) |
good | 2 | \( 1 - 2.07iT - 2T^{2} \) |
| 3 | \( 1 + 0.0823T + 3T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 7 | \( 1 + 3.12iT - 7T^{2} \) |
| 11 | \( 1 + 2.26iT - 11T^{2} \) |
| 13 | \( 1 + 0.833iT - 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 + 6.11T + 19T^{2} \) |
| 23 | \( 1 + 0.945T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 + 8.90T + 31T^{2} \) |
| 37 | \( 1 + 5.62T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 - 4.18iT - 43T^{2} \) |
| 47 | \( 1 - 3.06iT - 47T^{2} \) |
| 53 | \( 1 - 2.14iT - 53T^{2} \) |
| 59 | \( 1 - 4.18iT - 59T^{2} \) |
| 61 | \( 1 + 6.24iT - 61T^{2} \) |
| 67 | \( 1 - 2.56T + 67T^{2} \) |
| 71 | \( 1 + 9.12iT - 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 + 13.7iT - 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 14.9iT - 89T^{2} \) |
| 97 | \( 1 + 16.7iT - 97T^{2} \) |
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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
−11.21418753336347368074620171923, −10.57262773442761033497404054533, −8.900014228630686370563061161983, −8.130212154409640436206211378014, −7.60217454986677436360334000793, −6.65260982125399066815190389211, −5.62752603734473129922588175297, −4.41850518166900655544538718524, −3.39655760162130065533904469308, −0.12872763899312489647202001213,
2.12346357779551229309027467486, 3.24837517718127739416003051215, 4.23282706863082782455757417136, 5.51263827994889742249058940182, 7.00678014366164741858640614967, 8.360593707444269584248610657084, 8.900211831148348203398363611697, 10.08886673934794926296710632236, 11.05991448282201509830718489953, 11.72358668336376071681508455383