| L(s) = 1 | + 2.23·5-s − 7.74i·7-s + 6.92i·11-s + 22·13-s − 6.70·17-s − 27.1i·19-s − 29.4i·23-s + 5.00·25-s − 40.2·29-s + 19.3i·31-s − 17.3i·35-s + 2·37-s + 53.6·41-s − 15.4i·43-s + 13.8i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.10i·7-s + 0.629i·11-s + 1.69·13-s − 0.394·17-s − 1.42i·19-s − 1.28i·23-s + 0.200·25-s − 1.38·29-s + 0.624i·31-s − 0.494i·35-s + 0.0540·37-s + 1.30·41-s − 0.360i·43-s + 0.294i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.097950536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.097950536\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 + 7.74iT - 49T^{2} \) |
| 11 | \( 1 - 6.92iT - 121T^{2} \) |
| 13 | \( 1 - 22T + 169T^{2} \) |
| 17 | \( 1 + 6.70T + 289T^{2} \) |
| 19 | \( 1 + 27.1iT - 361T^{2} \) |
| 23 | \( 1 + 29.4iT - 529T^{2} \) |
| 29 | \( 1 + 40.2T + 841T^{2} \) |
| 31 | \( 1 - 19.3iT - 961T^{2} \) |
| 37 | \( 1 - 2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 53.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 15.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 13.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 6.70T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31T + 3.72e3T^{2} \) |
| 67 | \( 1 + 23.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 110. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 76T + 5.32e3T^{2} \) |
| 79 | \( 1 + 19.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 129. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 53.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + 32T + 9.40e3T^{2} \) |
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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
−8.830173435745243727517795151459, −7.903207198052495371726304631950, −6.99047278180165918126537386472, −6.53081256663946218857263983431, −5.55999800126947305266471731331, −4.50289393042389497749177703367, −3.92465945451352093478958912425, −2.76823742969333761932864280262, −1.59680169695039543470418274483, −0.54245034668820873482215305507,
1.25561480993411243329743980707, 2.16377751141339772657054805243, 3.34257168041660878304482728432, 4.03423783065252859800427068087, 5.59331053482706433376081585094, 5.72279026492969566119567566357, 6.48531317644886442151920960884, 7.75258800922781567141771680335, 8.344773695375295859534147041636, 9.175919213427991534650467254371
