L(s) = 1 | + (−0.717 − 1.21i)2-s + (−1.70 + 0.298i)3-s + (−0.971 + 1.74i)4-s + (−0.337 − 0.195i)5-s + (1.58 + 1.86i)6-s + (1.39 + 2.24i)7-s + (2.82 − 0.0704i)8-s + (2.82 − 1.01i)9-s + (0.00458 + 0.551i)10-s + (0.748 + 1.29i)11-s + (1.13 − 3.27i)12-s + 3.28·13-s + (1.73 − 3.31i)14-s + (0.634 + 0.232i)15-s + (−2.11 − 3.39i)16-s + (1.68 + 2.91i)17-s + ⋯ |
L(s) = 1 | + (−0.507 − 0.861i)2-s + (−0.985 + 0.172i)3-s + (−0.485 + 0.874i)4-s + (−0.151 − 0.0872i)5-s + (0.648 + 0.761i)6-s + (0.527 + 0.849i)7-s + (0.999 − 0.0249i)8-s + (0.940 − 0.339i)9-s + (0.00144 + 0.174i)10-s + (0.225 + 0.390i)11-s + (0.327 − 0.944i)12-s + 0.911·13-s + (0.464 − 0.885i)14-s + (0.163 + 0.0599i)15-s + (−0.528 − 0.848i)16-s + (0.407 + 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.664976 + 0.0343238i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.664976 + 0.0343238i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.717 + 1.21i)T \) |
| 3 | \( 1 + (1.70 - 0.298i)T \) |
| 7 | \( 1 + (-1.39 - 2.24i)T \) |
good | 5 | \( 1 + (0.337 + 0.195i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.748 - 1.29i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 + (-1.68 - 2.91i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.56 - 4.43i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.72 - 2.72i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 + (3.60 - 2.07i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.46 - 4.31i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 4.79iT - 43T^{2} \) |
| 47 | \( 1 + (-2.51 + 4.34i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.499 - 0.864i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.36 - 0.785i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.40 + 5.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.05 + 1.76i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.3iT - 71T^{2} \) |
| 73 | \( 1 + (-2.76 + 1.59i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.239 - 0.414i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 17.4iT - 83T^{2} \) |
| 89 | \( 1 + (2.54 - 4.41i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.00iT - 97T^{2} \) |
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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
−12.31527626664874980570604868187, −11.84160304078136917418903246231, −10.86392954440857818498906186993, −10.07407845236782371137751570671, −8.877502414929634347340475378566, −7.926629982766824190912419071242, −6.34984069846076668684867046969, −5.02281358765731706418371080211, −3.75258944043369683701359388557, −1.61964174772963645347365094880,
0.979388592477560173588574057262, 4.28245407452330878392702708206, 5.38687221665507261993919105345, 6.60109236989831388040168529847, 7.34687256548541965583237528440, 8.501796425329862412283347886846, 9.772149712441124399969049429062, 10.96871718621475652611154892147, 11.28773515028738792138788270052, 13.01726181551903858714753505710