| L(s) = 1 | + (−0.728 − 1.86i)2-s + (0.00678 + 0.00678i)3-s + (−2.93 + 2.71i)4-s + (0.00770 − 0.0175i)6-s + (7.19 + 3.50i)8-s − 8.99i·9-s + (−0.0383 − 0.00153i)12-s + (3.27 + 3.27i)13-s + (1.28 − 15.9i)16-s + (−16.7 + 6.55i)18-s − 23i·23-s + (0.0250 + 0.0726i)24-s − 25i·25-s + (3.71 − 8.47i)26-s + (0.122 − 0.122i)27-s + ⋯ |
| L(s) = 1 | + (−0.364 − 0.931i)2-s + (0.00226 + 0.00226i)3-s + (−0.734 + 0.678i)4-s + (0.00128 − 0.00293i)6-s + (0.899 + 0.437i)8-s − 0.999i·9-s + (−0.00319 − 0.000128i)12-s + (0.251 + 0.251i)13-s + (0.0800 − 0.996i)16-s + (−0.931 + 0.364i)18-s − i·23-s + (0.00104 + 0.00302i)24-s − i·25-s + (0.142 − 0.325i)26-s + (0.00452 − 0.00452i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.132781 - 0.842760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132781 - 0.842760i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.728 + 1.86i)T \) |
| 23 | \( 1 + 23iT \) |
| good | 3 | \( 1 + (-0.00678 - 0.00678i)T + 9iT^{2} \) |
| 5 | \( 1 + 25iT^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 + 121iT^{2} \) |
| 13 | \( 1 + (-3.27 - 3.27i)T + 169iT^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361iT^{2} \) |
| 29 | \( 1 + (36.2 + 36.2i)T + 841iT^{2} \) |
| 31 | \( 1 + 51.8T + 961T^{2} \) |
| 37 | \( 1 + 1.36e3iT^{2} \) |
| 41 | \( 1 + 8.78iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3iT^{2} \) |
| 47 | \( 1 - 42.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-44.5 + 44.5i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 - 3.72e3iT^{2} \) |
| 67 | \( 1 - 4.48e3iT^{2} \) |
| 71 | \( 1 - 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 144. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3iT^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−10.83933220852316026770938039129, −9.839987957469873064661826933432, −9.122117758242202767307871452986, −8.267635093341308762281523634848, −7.11528925003815758289095466099, −5.88117653746887764610100338028, −4.39291409171331944195554050977, −3.48816406087499329595938971174, −2.08621260449198626494439863526, −0.44185587396232691112696719233,
1.64535015130354673555772763106, 3.67172088379112295241919353550, 5.11036854740999120126077268585, 5.71632338388653127831506381542, 7.14322305837592089405816120934, 7.67899073997896672060790541533, 8.772567453852322208817736514208, 9.536125088043512671442747652549, 10.63763217863027702816783783790, 11.26637033773503839114009256479
