L(s) = 1 | + (−0.628 − 2.93i)3-s + 6.51·5-s + 7.64·7-s + (−8.21 + 3.68i)9-s + 17.8·11-s + 14.4i·13-s + (−4.09 − 19.1i)15-s + 18.6i·17-s + 12.9i·19-s + (−4.80 − 22.4i)21-s − 28.1i·23-s + 17.4·25-s + (15.9 + 21.7i)27-s + 3.08·29-s − 32.1·31-s + ⋯ |
L(s) = 1 | + (−0.209 − 0.977i)3-s + 1.30·5-s + 1.09·7-s + (−0.912 + 0.409i)9-s + 1.62·11-s + 1.10i·13-s + (−0.272 − 1.27i)15-s + 1.09i·17-s + 0.682i·19-s + (−0.228 − 1.06i)21-s − 1.22i·23-s + 0.696·25-s + (0.591 + 0.806i)27-s + 0.106·29-s − 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.20775 - 0.652199i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.20775 - 0.652199i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.628 + 2.93i)T \) |
good | 5 | \( 1 - 6.51T + 25T^{2} \) |
| 7 | \( 1 - 7.64T + 49T^{2} \) |
| 11 | \( 1 - 17.8T + 121T^{2} \) |
| 13 | \( 1 - 14.4iT - 169T^{2} \) |
| 17 | \( 1 - 18.6iT - 289T^{2} \) |
| 19 | \( 1 - 12.9iT - 361T^{2} \) |
| 23 | \( 1 + 28.1iT - 529T^{2} \) |
| 29 | \( 1 - 3.08T + 841T^{2} \) |
| 31 | \( 1 + 32.1T + 961T^{2} \) |
| 37 | \( 1 - 1.57iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 22.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 86.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 25.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 11.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 91.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 17.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 58.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 104.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 123.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 81.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 142. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 58.1T + 9.40e3T^{2} \) |
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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
−11.21112138704395172849535756630, −10.18854958850330941444059804331, −8.986330425209904887443728574827, −8.410347141873054243771269176267, −7.01246309439700425181290602189, −6.32799821632116653924761731043, −5.47329970110826309122120643542, −4.08955594327913520616367824370, −1.98121474526673868422548457594, −1.52167160877160378063390896118,
1.36500573372425292324519070883, 2.98634087868091521076189233243, 4.42695088368202848872517468369, 5.36262705944856710126625532136, 6.08958963109961603932295251188, 7.45287008603003103717270680128, 8.869150765847431140428259540529, 9.388457735507269197780777508895, 10.19162653164680520402772590652, 11.24485275710171467854886889823