L(s) = 1 | + (−0.956 + 1.04i)2-s + (−0.169 − 1.99i)4-s + (1.93 − 1.11i)5-s + 1.78i·7-s + (2.23 + 1.72i)8-s + (−0.697 + 3.08i)10-s + (3.34 − 3.34i)11-s + (−2.90 + 2.90i)13-s + (−1.86 − 1.71i)14-s + (−3.94 + 0.676i)16-s + 4.56·17-s + (0.0693 − 0.0693i)19-s + (−2.54 − 3.67i)20-s + (0.284 + 6.68i)22-s − 1.07·23-s + ⋯ |
L(s) = 1 | + (−0.676 + 0.736i)2-s + (−0.0849 − 0.996i)4-s + (0.867 − 0.497i)5-s + 0.675i·7-s + (0.791 + 0.611i)8-s + (−0.220 + 0.975i)10-s + (1.00 − 1.00i)11-s + (−0.805 + 0.805i)13-s + (−0.497 − 0.457i)14-s + (−0.985 + 0.169i)16-s + 1.10·17-s + (0.0159 − 0.0159i)19-s + (−0.569 − 0.822i)20-s + (0.0606 + 1.42i)22-s − 0.224·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27755 + 0.304450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27755 + 0.304450i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.956 - 1.04i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
good | 7 | \( 1 - 1.78iT - 7T^{2} \) |
| 11 | \( 1 + (-3.34 + 3.34i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.90 - 2.90i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 + (-0.0693 + 0.0693i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.07T + 23T^{2} \) |
| 29 | \( 1 + (1.23 - 1.23i)T - 29iT^{2} \) |
| 31 | \( 1 + 8.56iT - 31T^{2} \) |
| 37 | \( 1 + (-7.62 - 7.62i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.91T + 41T^{2} \) |
| 43 | \( 1 + (2.35 - 2.35i)T - 43iT^{2} \) |
| 47 | \( 1 + 6.32iT - 47T^{2} \) |
| 53 | \( 1 + (-3.61 + 3.61i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.25 + 2.25i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4.29 - 4.29i)T + 61iT^{2} \) |
| 67 | \( 1 + (5.85 + 5.85i)T + 67iT^{2} \) |
| 71 | \( 1 - 15.1iT - 71T^{2} \) |
| 73 | \( 1 + 6.88T + 73T^{2} \) |
| 79 | \( 1 - 12.9iT - 79T^{2} \) |
| 83 | \( 1 + (6.88 - 6.88i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 - 4.56iT - 97T^{2} \) |
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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
−9.951376287862832845188582821341, −9.558454229323416473739302906740, −8.806837801409712296602515355742, −8.068077983109496075267498893742, −6.89781475262184311206569484717, −5.98549074367675583815695839959, −5.50069455340670653076098118612, −4.28040128111391795476113283946, −2.41986131852229095549470209632, −1.11901528166016527884025542857,
1.20797740541976914413290680580, 2.42850422818392270679785873748, 3.53911210001369722862135735439, 4.66791990882742066569075873954, 6.01036954294996573262875549442, 7.25459567646120269007468246165, 7.55225511736121386902954817269, 8.985517777420241089654889050581, 9.702553083187054911011717864748, 10.24306212453656305796088541185