| L(s) = 1 | + (1.30 − 1.14i)3-s + (−2.23 − 1.41i)7-s + (0.381 − 2.97i)9-s − 1.13i·11-s + 0.333i·13-s − 4.20·17-s + 1.95i·19-s + (−4.52 + 0.719i)21-s + 2.54i·23-s + (−2.90 − 4.30i)27-s − 8.49i·29-s − 5.11i·31-s + (−1.30 − 1.47i)33-s − 2.23·37-s + (0.381 + 0.434i)39-s + ⋯ |
| L(s) = 1 | + (0.750 − 0.660i)3-s + (−0.845 − 0.534i)7-s + (0.127 − 0.991i)9-s − 0.342i·11-s + 0.0925i·13-s − 1.02·17-s + 0.448i·19-s + (−0.987 + 0.156i)21-s + 0.529i·23-s + (−0.559 − 0.828i)27-s − 1.57i·29-s − 0.918i·31-s + (−0.226 − 0.257i)33-s − 0.367·37-s + (0.0611 + 0.0695i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.041187589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041187589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.30 + 1.14i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
| good | 11 | \( 1 + 1.13iT - 11T^{2} \) |
| 13 | \( 1 - 0.333iT - 13T^{2} \) |
| 17 | \( 1 + 4.20T + 17T^{2} \) |
| 19 | \( 1 - 1.95iT - 19T^{2} \) |
| 23 | \( 1 - 2.54iT - 23T^{2} \) |
| 29 | \( 1 + 8.49iT - 29T^{2} \) |
| 31 | \( 1 + 5.11iT - 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 + 0.527T + 43T^{2} \) |
| 47 | \( 1 + 7.42T + 47T^{2} \) |
| 53 | \( 1 - 8.22iT - 53T^{2} \) |
| 59 | \( 1 + 3.59T + 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 - 10.7iT - 71T^{2} \) |
| 73 | \( 1 - 1.41iT - 73T^{2} \) |
| 79 | \( 1 - 1.47T + 79T^{2} \) |
| 83 | \( 1 + 5.81T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 9.69iT - 97T^{2} \) |
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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
−8.662457043957059305333332985743, −7.952177819303967397434716139135, −7.21373030086141961672328032083, −6.46740707172971351204095643602, −5.87654043883431236272557924638, −4.41300305663007791281841264067, −3.64562653427307865386590278307, −2.78778117380935451637461718482, −1.72671266827040590539635879257, −0.30564773336567441793370285059,
1.88211536272839701291477373248, 2.88037497223375206323807620965, 3.54881228422352512148375556047, 4.65980952744232732851815440651, 5.26452163773520757920465562993, 6.50602919571286847976812543137, 7.05541488077840408706523009436, 8.159321426625591517116109972108, 8.952155069070821465447205740106, 9.193682684886679667894446807020
