L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.72 − 2.98i)5-s + (0.5 + 0.866i)7-s − 0.999·8-s + 3.44·10-s + (1 + 1.73i)11-s + (2.44 − 4.24i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 2·17-s + 7.44·19-s + (1.72 + 2.98i)20-s + (−0.999 + 1.73i)22-s + (−0.5 + 0.866i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.771 − 1.33i)5-s + (0.188 + 0.327i)7-s − 0.353·8-s + 1.09·10-s + (0.301 + 0.522i)11-s + (0.679 − 1.17i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.485·17-s + 1.70·19-s + (0.385 + 0.667i)20-s + (−0.213 + 0.369i)22-s + (−0.104 + 0.180i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82762 + 0.244902i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82762 + 0.244902i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.44 + 4.24i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.44 - 2.51i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.44 - 2.51i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.89 + 8.48i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.72 + 9.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.55 + 2.68i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 + (3.94 + 6.84i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 + (-3.44 - 5.97i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.71540247330407120103729404513, −10.30022199385050991965398868055, −9.312130545629422392898298483076, −8.653368983781879179961302391061, −7.73367092251083696214711874071, −6.46019671878303665795526071095, −5.32617692616098358284295301354, −4.97303954481632471836363213763, −3.36279797727910818697042417223, −1.44334279050637367970325691865,
1.73462855171427635585311649090, 3.02968964935376171483801971320, 4.05339457385563941244863266042, 5.56412698012461024194042146986, 6.46961497929798402872694088916, 7.31063589104178823021623345883, 8.861027954349441165595628852636, 9.717150644119497236205012038067, 10.58021546615445608703081770980, 11.28720331658361211216692575756