| L(s) = 1 | + (1.14 + 1.30i)3-s + (−1.41 + 2.23i)7-s + (−0.381 + 2.97i)9-s − 1.13i·11-s − 0.333·13-s + 4.20i·17-s − 1.95i·19-s + (−4.52 + 0.719i)21-s − 2.54·23-s + (−4.30 + 2.90i)27-s + 8.49i·29-s − 5.11i·31-s + (1.47 − 1.30i)33-s + 2.23i·37-s + (−0.381 − 0.434i)39-s + ⋯ |
| L(s) = 1 | + (0.660 + 0.750i)3-s + (−0.534 + 0.845i)7-s + (−0.127 + 0.991i)9-s − 0.342i·11-s − 0.0925·13-s + 1.02i·17-s − 0.448i·19-s + (−0.987 + 0.156i)21-s − 0.529·23-s + (−0.828 + 0.559i)27-s + 1.57i·29-s − 0.918i·31-s + (0.257 − 0.226i)33-s + 0.367i·37-s + (−0.0611 − 0.0695i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.310230615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310230615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.14 - 1.30i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.41 - 2.23i)T \) |
| good | 11 | \( 1 + 1.13iT - 11T^{2} \) |
| 13 | \( 1 + 0.333T + 13T^{2} \) |
| 17 | \( 1 - 4.20iT - 17T^{2} \) |
| 19 | \( 1 + 1.95iT - 19T^{2} \) |
| 23 | \( 1 + 2.54T + 23T^{2} \) |
| 29 | \( 1 - 8.49iT - 29T^{2} \) |
| 31 | \( 1 + 5.11iT - 31T^{2} \) |
| 37 | \( 1 - 2.23iT - 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 + 0.527iT - 43T^{2} \) |
| 47 | \( 1 - 7.42iT - 47T^{2} \) |
| 53 | \( 1 + 8.22T + 53T^{2} \) |
| 59 | \( 1 - 3.59T + 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 6.70iT - 67T^{2} \) |
| 71 | \( 1 - 10.7iT - 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 + 1.47T + 79T^{2} \) |
| 83 | \( 1 + 5.81iT - 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 9.69T + 97T^{2} \) |
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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
−9.498485407077158118686695893489, −8.588145371040756072468460862571, −8.299151477856804515723786542695, −7.17970724390695599252698453922, −6.18841093425787087665122118279, −5.45278568238461850986135639994, −4.55043899822486527219159092757, −3.56250080869012391639490385647, −2.87473811212143715281392923428, −1.82558457525625264826089904949,
0.39751348776079812312291763400, 1.70240336770601719909401502969, 2.79354920188428626667317124105, 3.65231349035794264737017185965, 4.52748496434158811910489230117, 5.75842073009551727776263995219, 6.65655426037276377522860013849, 7.20760502143239221047873187831, 7.88721996984892544218303142544, 8.643300793654562309811660467377
