L(s) = 1 | + 0.151i·3-s − 3.45i·5-s + (2.64 + 0.100i)7-s + 2.97·9-s + (−2.52 + 2.14i)11-s + 13-s + 0.524·15-s − 4.73·17-s + 1.06·19-s + (−0.0151 + 0.401i)21-s − 8.86·23-s − 6.93·25-s + 0.907i·27-s − 0.523i·29-s − 7.62i·31-s + ⋯ |
L(s) = 1 | + 0.0876i·3-s − 1.54i·5-s + (0.999 + 0.0378i)7-s + 0.992·9-s + (−0.762 + 0.646i)11-s + 0.277·13-s + 0.135·15-s − 1.14·17-s + 0.243·19-s + (−0.00331 + 0.0875i)21-s − 1.84·23-s − 1.38·25-s + 0.174i·27-s − 0.0971i·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.331493631\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.331493631\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.64 - 0.100i)T \) |
| 11 | \( 1 + (2.52 - 2.14i)T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 0.151iT - 3T^{2} \) |
| 5 | \( 1 + 3.45iT - 5T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 + 8.86T + 23T^{2} \) |
| 29 | \( 1 + 0.523iT - 29T^{2} \) |
| 31 | \( 1 + 7.62iT - 31T^{2} \) |
| 37 | \( 1 + 5.23T + 37T^{2} \) |
| 41 | \( 1 + 0.443T + 41T^{2} \) |
| 43 | \( 1 + 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 6.76iT - 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 + 3.84iT - 59T^{2} \) |
| 61 | \( 1 + 6.13T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 6.85T + 71T^{2} \) |
| 73 | \( 1 - 8.18T + 73T^{2} \) |
| 79 | \( 1 + 5.60iT - 79T^{2} \) |
| 83 | \( 1 + 2.69T + 83T^{2} \) |
| 89 | \( 1 - 11.9iT - 89T^{2} \) |
| 97 | \( 1 - 6.19iT - 97T^{2} \) |
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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
−8.111245007748532032061966464015, −7.70529016055064368379121586380, −6.77683050246456368540058924662, −5.65337849403713854640831827270, −5.08097324841754524516986170765, −4.35290733956582721662520967568, −3.98406679963271690651339330116, −2.10586019938130780217598936497, −1.70845433696248247951720753885, −0.35630963680520834999928491199,
1.50473689233386800366341858900, 2.37052851050805731922872598091, 3.25006519026894451346382450235, 4.15075117349164184527311790663, 4.92143128346811775109619625555, 5.98527791566668836039240175722, 6.56099449861890760512613333326, 7.33018741138001612355725642450, 7.87047940895136239407948278013, 8.533413529949819631127832735126