| L(s) = 1 | + (−2.56 − 1.55i)3-s + 1.34·7-s + (4.17 + 7.97i)9-s + 3.66i·11-s + 7.34·13-s + 6.31i·17-s − 30.7·19-s + (−3.44 − 2.08i)21-s − 26.2i·23-s + (1.68 − 26.9i)27-s + 40.9i·29-s + 9.97·31-s + (5.69 − 9.40i)33-s + 39.4·37-s + (−18.8 − 11.4i)39-s + ⋯ |
| L(s) = 1 | + (−0.855 − 0.517i)3-s + 0.191·7-s + (0.463 + 0.886i)9-s + 0.333i·11-s + 0.564·13-s + 0.371i·17-s − 1.61·19-s + (−0.164 − 0.0993i)21-s − 1.14i·23-s + (0.0624 − 0.998i)27-s + 1.41i·29-s + 0.321·31-s + (0.172 − 0.285i)33-s + 1.06·37-s + (−0.483 − 0.292i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1485239108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1485239108\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.56 + 1.55i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.34T + 49T^{2} \) |
| 11 | \( 1 - 3.66iT - 121T^{2} \) |
| 13 | \( 1 - 7.34T + 169T^{2} \) |
| 17 | \( 1 - 6.31iT - 289T^{2} \) |
| 19 | \( 1 + 30.7T + 361T^{2} \) |
| 23 | \( 1 + 26.2iT - 529T^{2} \) |
| 29 | \( 1 - 40.9iT - 841T^{2} \) |
| 31 | \( 1 - 9.97T + 961T^{2} \) |
| 37 | \( 1 - 39.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 68.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 24.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 79.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 38.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 39.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 64.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 136. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 79.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 0.0506iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 22.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 35.0T + 9.40e3T^{2} \) |
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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
−10.30705833719937284758059155677, −8.861874141739950085200959623368, −8.308054927663879172964107031189, −7.23727925164032558980182688070, −6.54901214236669755958666974244, −5.82623676194713770054241409983, −4.81443176593796955835446987459, −4.01570930906479694028518682281, −2.42888527758613090424831345886, −1.37233133263630295324048497120,
0.05167165070690376153273612824, 1.46569496406580831212340553158, 3.04933916029684460970099807609, 4.19387590976822135139144584577, 4.81524221612021794606783816796, 6.08757511326520417445807397988, 6.28331626117347026719627681099, 7.60714943505047163185917775983, 8.424936865045660600531754914655, 9.438883567882163466812973314581
