| L(s) = 1 | − 3.04i·3-s + 4.30i·5-s + (2.63 + 0.225i)7-s − 6.29·9-s + (2.23 − 2.45i)11-s − 13-s + 13.1·15-s − 6.22·17-s − 0.815·19-s + (0.688 − 8.03i)21-s − 0.626·23-s − 13.5·25-s + 10.0i·27-s − 5.50i·29-s − 1.54i·31-s + ⋯ |
| L(s) = 1 | − 1.76i·3-s + 1.92i·5-s + (0.996 + 0.0854i)7-s − 2.09·9-s + (0.672 − 0.740i)11-s − 0.277·13-s + 3.38·15-s − 1.51·17-s − 0.187·19-s + (0.150 − 1.75i)21-s − 0.130·23-s − 2.70·25-s + 1.93i·27-s − 1.02i·29-s − 0.277i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.284781861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284781861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.63 - 0.225i)T \) |
| 11 | \( 1 + (-2.23 + 2.45i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.04iT - 3T^{2} \) |
| 5 | \( 1 - 4.30iT - 5T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 + 0.815T + 19T^{2} \) |
| 23 | \( 1 + 0.626T + 23T^{2} \) |
| 29 | \( 1 + 5.50iT - 29T^{2} \) |
| 31 | \( 1 + 1.54iT - 31T^{2} \) |
| 37 | \( 1 + 5.94T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 2.96iT - 43T^{2} \) |
| 47 | \( 1 + 10.6iT - 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 7.99iT - 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 - 6.46T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 9.52iT - 79T^{2} \) |
| 83 | \( 1 - 8.25T + 83T^{2} \) |
| 89 | \( 1 + 1.67iT - 89T^{2} \) |
| 97 | \( 1 + 3.51iT - 97T^{2} \) |
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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
−8.029706816911512451558032811319, −7.22930380380073290247894726014, −6.83117178506523933558163742772, −6.24288031292015170963983792711, −5.63079404088316371595443873858, −4.20938555981826691731282122910, −3.19748966334111975812534217304, −2.23466831230485121641250215113, −1.94281567365824774649314353659, −0.36619108458216914814445921164,
1.22799437879039784943864642799, 2.31197396677271483452334403453, 3.87940067128387109304753854155, 4.33171667439599668912239439265, 4.87090649825768807057024140130, 5.22104577380456603599462943250, 6.24336738475216246141097651810, 7.52218278229934512297388921778, 8.417446583774183724023356928388, 8.875302168592335927621529925758
