L(s) = 1 | + (1.36 + 0.366i)2-s + (−1.67 + 0.448i)3-s + (1.73 + i)4-s + (−4.95 − 0.641i)5-s − 2.44·6-s + (2.95 + 6.34i)7-s + (1.99 + 2i)8-s + (2.59 − 1.50i)9-s + (−6.53 − 2.69i)10-s + (−9.98 + 17.2i)11-s + (−3.34 − 0.896i)12-s + (−13.3 − 13.3i)13-s + (1.71 + 9.74i)14-s + (8.58 − 1.14i)15-s + (1.99 + 3.46i)16-s + (5.85 + 21.8i)17-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (−0.557 + 0.149i)3-s + (0.433 + 0.250i)4-s + (−0.991 − 0.128i)5-s − 0.408·6-s + (0.422 + 0.906i)7-s + (0.249 + 0.250i)8-s + (0.288 − 0.166i)9-s + (−0.653 − 0.269i)10-s + (−0.907 + 1.57i)11-s + (−0.278 − 0.0747i)12-s + (−1.02 − 1.02i)13-s + (0.122 + 0.696i)14-s + (0.572 − 0.0765i)15-s + (0.124 + 0.216i)16-s + (0.344 + 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.739i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.454459 + 1.02739i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.454459 + 1.02739i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 3 | \( 1 + (1.67 - 0.448i)T \) |
| 5 | \( 1 + (4.95 + 0.641i)T \) |
| 7 | \( 1 + (-2.95 - 6.34i)T \) |
good | 11 | \( 1 + (9.98 - 17.2i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (13.3 + 13.3i)T + 169iT^{2} \) |
| 17 | \( 1 + (-5.85 - 21.8i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (14.8 - 8.58i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (5.43 - 20.2i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 23.1iT - 841T^{2} \) |
| 31 | \( 1 + (-27.7 + 48.0i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-22.2 - 5.97i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 21.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-14.6 - 14.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-73.1 - 19.6i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-4.01 + 1.07i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (26.7 + 15.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.75 + 3.04i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-0.764 - 2.85i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 11.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-21.0 + 5.65i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-65.5 + 37.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-28.2 - 28.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (124. - 71.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-56.0 + 56.0i)T - 9.40e3iT^{2} \) |
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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
−12.40080019971682768377170903735, −11.90064158258300163668480418403, −10.72958331179610831124996465308, −9.793589258250209000812471398138, −8.016826198676991060876129434260, −7.62255963337145432029862053992, −6.02711419393269882905555636113, −5.03679041458380092353542563790, −4.15270699139456003635967349431, −2.38348763468158809267363592519,
0.52507656120272725444614665177, 2.88585994167144470027767108206, 4.33224625853716547702958289744, 5.13741334263275157924698697315, 6.70707802309942805351448147943, 7.43729502318055857294851132442, 8.614280453727753569690355423366, 10.37921725797669735774501725276, 11.00885160543671417943813691794, 11.76528878912712440477167064657