L(s) = 1 | + 2.86·5-s + (2.43 − 1.03i)7-s + 3.32i·11-s + 0.821i·13-s + 3.52·17-s + 5.25i·19-s + 5.12i·23-s + 3.22·25-s − 1.64i·29-s − 2.55i·31-s + (6.98 − 2.96i)35-s − 8.93·37-s + 3.49·41-s − 0.161·43-s + 5.34·47-s + ⋯ |
L(s) = 1 | + 1.28·5-s + (0.920 − 0.391i)7-s + 1.00i·11-s + 0.227i·13-s + 0.853·17-s + 1.20i·19-s + 1.06i·23-s + 0.645·25-s − 0.305i·29-s − 0.458i·31-s + (1.18 − 0.501i)35-s − 1.46·37-s + 0.546·41-s − 0.0245·43-s + 0.779·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.430702790\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.430702790\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.43 + 1.03i)T \) |
good | 5 | \( 1 - 2.86T + 5T^{2} \) |
| 11 | \( 1 - 3.32iT - 11T^{2} \) |
| 13 | \( 1 - 0.821iT - 13T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 - 5.25iT - 19T^{2} \) |
| 23 | \( 1 - 5.12iT - 23T^{2} \) |
| 29 | \( 1 + 1.64iT - 29T^{2} \) |
| 31 | \( 1 + 2.55iT - 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 + 0.161T + 43T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 + 3.28iT - 53T^{2} \) |
| 59 | \( 1 + 4.08T + 59T^{2} \) |
| 61 | \( 1 + 8.43iT - 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 5.68iT - 71T^{2} \) |
| 73 | \( 1 + 7.86iT - 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 12.8iT - 97T^{2} \) |
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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
−9.738855401545777927304281124988, −8.823954449410544859360060867690, −7.78209307404741606545960594870, −7.25899467548194270436462650168, −6.09146751653275840819234560400, −5.46248778347878208677566574365, −4.61555824190188058354615851726, −3.54279199883309385372885392291, −2.04689976877444678177254672938, −1.49446583611735210224831392716,
1.08795465231508797350559903380, 2.24915603918437418344975392197, 3.13866641565332157445987505325, 4.59659148445740702402656455101, 5.43267562871692643264800771050, 5.94808058502884280952591409844, 6.92144305408143960241507112455, 7.936699367452924463398721441084, 8.833441562729760988221152700003, 9.181462708772673610842000646507