L(s) = 1 | − 2i·2-s − 8i·3-s − 4·4-s − 16·6-s + 4i·7-s + 8i·8-s − 37·9-s + 12·11-s + 32i·12-s − 58i·13-s + 8·14-s + 16·16-s − 66i·17-s + 74i·18-s + 100·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.53i·3-s − 0.5·4-s − 1.08·6-s + 0.215i·7-s + 0.353i·8-s − 1.37·9-s + 0.328·11-s + 0.769i·12-s − 1.23i·13-s + 0.152·14-s + 0.250·16-s − 0.941i·17-s + 0.968i·18-s + 1.20·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.282444 - 1.19645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.282444 - 1.19645i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 8iT - 27T^{2} \) |
| 7 | \( 1 - 4iT - 343T^{2} \) |
| 11 | \( 1 - 12T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 100T + 6.85e3T^{2} \) |
| 23 | \( 1 - 132iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 152T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 438T + 6.89e4T^{2} \) |
| 43 | \( 1 - 32iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 204iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 222iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 420T + 2.05e5T^{2} \) |
| 61 | \( 1 - 902T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.02e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 432T + 3.57e5T^{2} \) |
| 73 | \( 1 - 362iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 160T + 4.93e5T^{2} \) |
| 83 | \( 1 - 72iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 810T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3iT - 9.12e5T^{2} \) |
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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
−14.00096104293718566486892224623, −13.31271593880255605271715297694, −12.21457394514825895843208794355, −11.51417460820323294099251633669, −9.832569946823623660128707259755, −8.288851680221354422659785850911, −7.15420894931494698984089348467, −5.50750450250675981569748446203, −2.90361989337505064412224784537, −1.05760840811300091664909075802,
3.82831637849933163936845337038, 4.96052845392361014241777925406, 6.60120719230790750030694868445, 8.471655985474746939248355619300, 9.539024558288012235209398976354, 10.52235850261201176709979784220, 11.92747077283982877643766860601, 13.78049441861810918113040198402, 14.66146576740618367689496819192, 15.62001423636497566063866823441