L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s − 13-s + (0.866 + 0.5i)19-s + 0.999·21-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + (0.866 + 1.5i)31-s + (−1.5 − 0.866i)37-s + (−0.866 + 0.5i)39-s − 1.73·43-s + (0.499 + 0.866i)49-s + 0.999·57-s + (1 − 1.73i)61-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s − 13-s + (0.866 + 0.5i)19-s + 0.999·21-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + (0.866 + 1.5i)31-s + (−1.5 − 0.866i)37-s + (−0.866 + 0.5i)39-s − 1.73·43-s + (0.499 + 0.866i)49-s + 0.999·57-s + (1 − 1.73i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.526108534\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526108534\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.748536181835068789744610098884, −8.682475026560644773055278783217, −8.275084870386044187464571722951, −7.40738004030981844571667392295, −6.72504400094645017120915150762, −5.51571160790512452814863516940, −4.69584769247894584081691715986, −3.50709903302913664838614361309, −2.49028137727555624788774596405, −1.54834316803383641595104936996,
1.66572134798364565145147706325, 2.78913278998542095954167168888, 3.82119273590295522036583662870, 4.75490889786588097918177324479, 5.34504428914436694972556872303, 6.88891587410354316156235346640, 7.58957581643651051864609489592, 8.195639030600138105890559913328, 9.081048090149080470155905231073, 9.886832878486278746608135667211