L(s) = 1 | − i·3-s + i·7-s − 9-s − 4·11-s + 3i·13-s − 4i·17-s + 19-s + 21-s + i·27-s + 8·29-s + 31-s + 4i·33-s + 2i·37-s + 3·39-s + 2·41-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.20·11-s + 0.832i·13-s − 0.970i·17-s + 0.229·19-s + 0.218·21-s + 0.192i·27-s + 1.48·29-s + 0.179·31-s + 0.696i·33-s + 0.328i·37-s + 0.480·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.435124514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.435124514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 11iT - 97T^{2} \) |
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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
−8.863771024278024627842282771767, −8.260092773179904227575010232715, −7.43352426691261945998355987388, −6.81525462667645190425728318182, −5.91904881863551784141706248904, −5.13555495728022412626978133650, −4.33400306114065246649581609742, −2.91565845745533697559344947051, −2.39314886131597067972497068546, −0.998130716069444828159227634325,
0.58503994907227031916759891316, 2.23442900038762032431635233913, 3.20362783544027521640094087752, 4.03528920200095265461536978678, 5.02606133232270442284528364130, 5.59003678178637576274541317285, 6.54209096557011535309008619499, 7.52265549617193417269822822219, 8.201511441752691545701327914464, 8.780724800611461752648206592333