L(s) = 1 | + (−1.21 − 2.10i)5-s + (1.05 − 2.42i)7-s + (−2.09 − 1.21i)11-s + (−4.73 + 2.73i)13-s − 2.58·17-s − 0.402i·19-s + (−3.06 + 1.77i)23-s + (−0.440 + 0.762i)25-s + (6.31 + 3.64i)29-s + (−3.63 + 2.10i)31-s + (−6.37 + 0.722i)35-s − 3.19·37-s + (−4.03 − 6.99i)41-s + (−4.22 + 7.31i)43-s + (2.25 − 3.91i)47-s + ⋯ |
L(s) = 1 | + (−0.542 − 0.939i)5-s + (0.399 − 0.916i)7-s + (−0.632 − 0.365i)11-s + (−1.31 + 0.758i)13-s − 0.625·17-s − 0.0923i·19-s + (−0.639 + 0.369i)23-s + (−0.0880 + 0.152i)25-s + (1.17 + 0.677i)29-s + (−0.653 + 0.377i)31-s + (−1.07 + 0.122i)35-s − 0.525·37-s + (−0.630 − 1.09i)41-s + (−0.644 + 1.11i)43-s + (0.329 − 0.570i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0735583 - 0.575604i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0735583 - 0.575604i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.05 + 2.42i)T \) |
good | 5 | \( 1 + (1.21 + 2.10i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.09 + 1.21i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.73 - 2.73i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 0.402iT - 19T^{2} \) |
| 23 | \( 1 + (3.06 - 1.77i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.31 - 3.64i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.63 - 2.10i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 + (4.03 + 6.99i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.22 - 7.31i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.25 + 3.91i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 14.0iT - 53T^{2} \) |
| 59 | \( 1 + (-0.0779 - 0.134i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.2 + 5.90i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.53 - 4.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.73iT - 71T^{2} \) |
| 73 | \( 1 + 8.80iT - 73T^{2} \) |
| 79 | \( 1 + (-5.66 + 9.81i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.50 - 13.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + (-4.97 - 2.87i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.01326844357957207936684156095, −8.982279448052412430067434564622, −8.227704837877888925528881930397, −7.44573034570639130444761124884, −6.61445676597130906194674196240, −5.03223110656383446386770734944, −4.68214756424836130595720107321, −3.52034236775033604097377920372, −1.90510621121031195066786325305, −0.27683857876404816363124707995,
2.28258998418141117723089663250, 2.98928105168353667084996002015, 4.45233337163088638548536063942, 5.34549565317451534285257680666, 6.40538577539030798321634318865, 7.43116113802940638443750016605, 7.982980580454076227390279323871, 8.993444275606153214625521271864, 10.09268599629593730530637484340, 10.62064961572271332692155604910