L(s) = 1 | + 1.39·3-s + (−2.17 + 0.535i)5-s + (−2.13 + 2.13i)7-s − 1.05·9-s + (−2.17 − 2.17i)11-s − 1.54i·13-s + (−3.02 + 0.745i)15-s + (−3.86 + 3.86i)17-s + (−0.0136 − 0.0136i)19-s + (−2.97 + 2.97i)21-s + (−3.15 − 3.15i)23-s + (4.42 − 2.32i)25-s − 5.65·27-s + (−3.33 + 3.33i)29-s − 8.92i·31-s + ⋯ |
L(s) = 1 | + 0.804·3-s + (−0.970 + 0.239i)5-s + (−0.806 + 0.806i)7-s − 0.353·9-s + (−0.654 − 0.654i)11-s − 0.428i·13-s + (−0.780 + 0.192i)15-s + (−0.937 + 0.937i)17-s + (−0.00313 − 0.00313i)19-s + (−0.648 + 0.648i)21-s + (−0.657 − 0.657i)23-s + (0.885 − 0.464i)25-s − 1.08·27-s + (−0.619 + 0.619i)29-s − 1.60i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0172584 + 0.250364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0172584 + 0.250364i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.17 - 0.535i)T \) |
good | 3 | \( 1 - 1.39T + 3T^{2} \) |
| 7 | \( 1 + (2.13 - 2.13i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.17 + 2.17i)T + 11iT^{2} \) |
| 13 | \( 1 + 1.54iT - 13T^{2} \) |
| 17 | \( 1 + (3.86 - 3.86i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.0136 + 0.0136i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.15 + 3.15i)T + 23iT^{2} \) |
| 29 | \( 1 + (3.33 - 3.33i)T - 29iT^{2} \) |
| 31 | \( 1 + 8.92iT - 31T^{2} \) |
| 37 | \( 1 - 7.24iT - 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 2.02iT - 43T^{2} \) |
| 47 | \( 1 + (-3.34 - 3.34i)T + 47iT^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 + (-3.52 + 3.52i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.41 + 1.41i)T + 61iT^{2} \) |
| 67 | \( 1 - 0.748iT - 67T^{2} \) |
| 71 | \( 1 + 0.269T + 71T^{2} \) |
| 73 | \( 1 + (-0.811 + 0.811i)T - 73iT^{2} \) |
| 79 | \( 1 + 2.80T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + (-6.33 + 6.33i)T - 97iT^{2} \) |
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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
−11.06458393292466645317904025248, −10.06566460876696402145729814906, −9.026643173792531895202674187814, −8.329052008610220799311189828787, −7.83129443104884092601535303661, −6.49236314949365398895014305914, −5.70960535731852953584013770592, −4.22538098633987307452226946198, −3.18708020432134633776488530506, −2.50628399963496160173664956976,
0.11313361502350011823030179366, 2.33532756461185783855997400737, 3.51839698509370798793745985475, 4.23986600506811189635023098553, 5.48590704669803408939327647864, 7.12257944606103505688854398793, 7.31355438699805375583304160098, 8.507094449670282706311616087586, 9.143515289472638495285969575286, 10.07444473856430963093577023287