L(s) = 1 | + 0.571i·2-s − i·3-s + 1.67·4-s + 0.571·6-s + 1.42i·7-s + 2.10i·8-s − 9-s + 3.24·11-s − 1.67i·12-s − i·13-s − 0.816·14-s + 2.14·16-s + 1.85i·17-s − 0.571i·18-s + 1.81·19-s + ⋯ |
L(s) = 1 | + 0.404i·2-s − 0.577i·3-s + 0.836·4-s + 0.233·6-s + 0.539i·7-s + 0.742i·8-s − 0.333·9-s + 0.978·11-s − 0.482i·12-s − 0.277i·13-s − 0.218·14-s + 0.535·16-s + 0.450i·17-s − 0.134i·18-s + 0.416·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.03911 + 0.481369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03911 + 0.481369i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 - 0.571iT - 2T^{2} \) |
| 7 | \( 1 - 1.42iT - 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 17 | \( 1 - 1.85iT - 17T^{2} \) |
| 19 | \( 1 - 1.81T + 19T^{2} \) |
| 23 | \( 1 - 1.52iT - 23T^{2} \) |
| 29 | \( 1 + 2.34T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 3.52iT - 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 + 10.0iT - 43T^{2} \) |
| 47 | \( 1 - 11.2iT - 47T^{2} \) |
| 53 | \( 1 + 6.81iT - 53T^{2} \) |
| 59 | \( 1 - 5.91T + 59T^{2} \) |
| 61 | \( 1 - 5.48T + 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 9.81T + 71T^{2} \) |
| 73 | \( 1 + 5.32iT - 73T^{2} \) |
| 79 | \( 1 - 2.96T + 79T^{2} \) |
| 83 | \( 1 + 3.14iT - 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 - 1.51iT - 97T^{2} \) |
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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
−10.08932111293048838132317010449, −9.033352703147176805330207870242, −8.250817509673524691321594265795, −7.46717136874755700162035282537, −6.61865104457482719747928520138, −6.03230847919906206532415135147, −5.13194639966016353169924075034, −3.61214173654488836065981682241, −2.49551731674226799527677491131, −1.40161473094106842375430269217,
1.14156657728844258813537225824, 2.54028078774760268499247371892, 3.61277829827314489192312517885, 4.38554711460991127668667991895, 5.64905931520331832370922769751, 6.64943086646675074833544723145, 7.25068968151920138871124285160, 8.370573363103408041824950722523, 9.382086198047835116542690149953, 10.03810576247942220534989730191