L(s) = 1 | − 1.41i·2-s + (2.85 + 0.922i)3-s − 2.00·4-s + 2.23i·5-s + (1.30 − 4.03i)6-s + 2.64·7-s + 2.82i·8-s + (7.29 + 5.26i)9-s + 3.16·10-s − 4.58i·11-s + (−5.70 − 1.84i)12-s + 20.4·13-s − 3.74i·14-s + (−2.06 + 6.38i)15-s + 4.00·16-s + 4.97i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.951 + 0.307i)3-s − 0.500·4-s + 0.447i·5-s + (0.217 − 0.672i)6-s + 0.377·7-s + 0.353i·8-s + (0.810 + 0.585i)9-s + 0.316·10-s − 0.416i·11-s + (−0.475 − 0.153i)12-s + 1.57·13-s − 0.267i·14-s + (−0.137 + 0.425i)15-s + 0.250·16-s + 0.292i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.12012 - 0.334150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12012 - 0.334150i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-2.85 - 0.922i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
good | 11 | \( 1 + 4.58iT - 121T^{2} \) |
| 13 | \( 1 - 20.4T + 169T^{2} \) |
| 17 | \( 1 - 4.97iT - 289T^{2} \) |
| 19 | \( 1 - 6.22T + 361T^{2} \) |
| 23 | \( 1 - 0.140iT - 529T^{2} \) |
| 29 | \( 1 - 1.76iT - 841T^{2} \) |
| 31 | \( 1 + 29.7T + 961T^{2} \) |
| 37 | \( 1 + 38.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 68.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 57.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 41.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 79.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 19.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 67.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 30.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 111.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 156. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 133. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 20.1T + 9.40e3T^{2} \) |
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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
−11.99952621434448165099290675491, −10.87270320907476812080109621767, −10.37197939758240264321021335586, −9.020817568620318980912287825094, −8.468331368045343386945468247683, −7.24325804084078518695274179425, −5.63484183241193961121285641752, −4.05236727844919101508837552987, −3.20364381914309351141482512921, −1.68427044924047643697294176097,
1.46759390101367870892409673974, 3.45782537927774142727587528075, 4.69867472149059654854815322174, 6.12279039461151526820928501925, 7.28007303539870393231071354891, 8.224093325358332421057999690272, 8.909736366138159145861789266051, 9.864852899107664297372132200288, 11.26948791532448300004379374436, 12.55422417360647279682423190088