L(s) = 1 | + (1.03 − 1.03i)3-s + (−1.21 − 1.87i)5-s + (1.58 − 2.11i)7-s + 0.841i·9-s − 2.34·11-s + (1.96 − 1.96i)13-s + (−3.21 − 0.679i)15-s + (−5.15 − 5.15i)17-s + 3.74·19-s + (−0.547 − 3.84i)21-s + (6.08 + 6.08i)23-s + (−2.02 + 4.57i)25-s + (3.99 + 3.99i)27-s − 5.89i·29-s + 1.56i·31-s + ⋯ |
L(s) = 1 | + (0.599 − 0.599i)3-s + (−0.545 − 0.838i)5-s + (0.600 − 0.799i)7-s + 0.280i·9-s − 0.707·11-s + (0.544 − 0.544i)13-s + (−0.829 − 0.175i)15-s + (−1.25 − 1.25i)17-s + 0.859·19-s + (−0.119 − 0.839i)21-s + (1.26 + 1.26i)23-s + (−0.404 + 0.914i)25-s + (0.768 + 0.768i)27-s − 1.09i·29-s + 0.281i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08940 - 0.923347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08940 - 0.923347i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.21 + 1.87i)T \) |
| 7 | \( 1 + (-1.58 + 2.11i)T \) |
good | 3 | \( 1 + (-1.03 + 1.03i)T - 3iT^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 + (-1.96 + 1.96i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.15 + 5.15i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 + (-6.08 - 6.08i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.89iT - 29T^{2} \) |
| 31 | \( 1 - 1.56iT - 31T^{2} \) |
| 37 | \( 1 + (-1.53 + 1.53i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.51iT - 41T^{2} \) |
| 43 | \( 1 + (-1.86 - 1.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.59 - 4.59i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.88 - 3.88i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.62T + 59T^{2} \) |
| 61 | \( 1 - 2.00iT - 61T^{2} \) |
| 67 | \( 1 + (-2.69 + 2.69i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.392T + 71T^{2} \) |
| 73 | \( 1 + (7.08 - 7.08i)T - 73iT^{2} \) |
| 79 | \( 1 - 4.98iT - 79T^{2} \) |
| 83 | \( 1 + (9.36 - 9.36i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 + (-11.8 - 11.8i)T + 97iT^{2} \) |
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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
−11.50362431663289334145126846853, −10.99584498890399020496926170443, −9.566655311973993374209984009003, −8.555975391615726213772106531906, −7.71376215994080109362392613040, −7.22776813652062655922340654705, −5.34673843858752142328424254498, −4.45480683680093081339306400637, −2.89311440286054397651391383380, −1.14651663882308992872207997871,
2.41710202413787496296632982285, 3.57492327187724603200178892606, 4.69762813692824843443275091604, 6.16121285876203241799486342545, 7.24090491974367396925050644320, 8.590151768641324142711773822597, 8.885684121267205773285188702686, 10.37051553242129520319096612560, 10.98293378013688574178541582643, 11.92869382413988206007313402625