L(s) = 1 | − i·3-s − 2.23i·5-s + 2i·7-s − 9-s + 4.47·11-s − 4.47i·13-s − 2.23·15-s − 4.47i·17-s + 2·21-s − 4i·23-s − 5.00·25-s + i·27-s − 4·29-s − 8.94·31-s − 4.47i·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.999i·5-s + 0.755i·7-s − 0.333·9-s + 1.34·11-s − 1.24i·13-s − 0.577·15-s − 1.08i·17-s + 0.436·21-s − 0.834i·23-s − 1.00·25-s + 0.192i·27-s − 0.742·29-s − 1.60·31-s − 0.778i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.977330 - 0.977330i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.977330 - 0.977330i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.23iT \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 4.47iT - 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 8.94iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 17.8iT - 97T^{2} \) |
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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
−11.00780533410149020559346495720, −9.499809555280150232250857824676, −9.048042519129106441992261455646, −8.120238374412916588852118791251, −7.17481129996176074466013657054, −5.94656397782954488739792514355, −5.26420782644790667731134628201, −3.92778318368704210014756033019, −2.41395148949177735584596017201, −0.902769540828003463034430719995,
1.87457240288348000810015451236, 3.75372778525713033668020913087, 3.98415373856896940226694788275, 5.68707161284699145645261759313, 6.71939582125705824043611505356, 7.33401191347887864120452412058, 8.712404386087813586316064678506, 9.557638982794712752143434663881, 10.31864358410556232490009895903, 11.27373005236905155295000423311