L(s) = 1 | + (−1.41 − i)3-s + (−1 + 2i)5-s + (−0.414 + 0.414i)7-s + (1.00 + 2.82i)9-s + 4.82i·11-s + (−1.82 − 1.82i)13-s + (3.41 − 1.82i)15-s + (3.82 + 3.82i)17-s + 4.82i·19-s + (1 − 0.171i)21-s + (−1.58 + 1.58i)23-s + (−3 − 4i)25-s + (1.41 − 5.00i)27-s − 7.65·29-s + 5.65·31-s + ⋯ |
L(s) = 1 | + (−0.816 − 0.577i)3-s + (−0.447 + 0.894i)5-s + (−0.156 + 0.156i)7-s + (0.333 + 0.942i)9-s + 1.45i·11-s + (−0.507 − 0.507i)13-s + (0.881 − 0.472i)15-s + (0.928 + 0.928i)17-s + 1.10i·19-s + (0.218 − 0.0374i)21-s + (−0.330 + 0.330i)23-s + (−0.600 − 0.800i)25-s + (0.272 − 0.962i)27-s − 1.42·29-s + 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.501033 + 0.470938i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.501033 + 0.470938i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 + i)T \) |
| 5 | \( 1 + (1 - 2i)T \) |
good | 7 | \( 1 + (0.414 - 0.414i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 13 | \( 1 + (1.82 + 1.82i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.82 - 3.82i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.82iT - 19T^{2} \) |
| 23 | \( 1 + (1.58 - 1.58i)T - 23iT^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (0.171 - 0.171i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (2.41 + 2.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.41 + 6.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (3 - 3i)T - 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + (-4.07 + 4.07i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.48iT - 71T^{2} \) |
| 73 | \( 1 + (-6.65 - 6.65i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.82iT - 79T^{2} \) |
| 83 | \( 1 + (-5.24 + 5.24i)T - 83iT^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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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
−12.28643510206877428490172570474, −11.59350415911250702280339396890, −10.35615438429173962982686885209, −9.951563273913570208722849070977, −7.978234756586393929514733428879, −7.39269586426901635017273492407, −6.36679599390417215470791497846, −5.30956871714691693882438195892, −3.80945201115467149013523338902, −2.03791085785182109504312541004,
0.61869269536148123988064184944, 3.40156015935742667575846942575, 4.66252222966729972445850437089, 5.50555446597866406431859449225, 6.71771889078974691931238367057, 8.046532119129929370378495464367, 9.162944335905438543392380625398, 9.891017560993633487925871936998, 11.32135150666001708949962608710, 11.57864446298623278250321512176