L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−1 + 1.73i)13-s + 0.999·15-s + 2·17-s + (−0.499 + 0.866i)21-s + (−0.499 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)29-s + (0.499 − 0.866i)33-s + 0.999·35-s + 1.99·39-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−1 + 1.73i)13-s + 0.999·15-s + 2·17-s + (−0.499 + 0.866i)21-s + (−0.499 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)29-s + (0.499 − 0.866i)33-s + 0.999·35-s + 1.99·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7048396350\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7048396350\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 2T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−10.06730644495767293612875970793, −9.348551761298826515776756762890, −7.959218325176305146960312070330, −7.20111279648771999587577018523, −6.98078418398655709014225964889, −6.12070785493283817583891144665, −4.83677616947501216250468290792, −3.91042395338467591995970446413, −2.75524952987346086353900111449, −1.45142171725875049176399491500,
0.70555636344158107069093094353, 2.97895615242959328264702728955, 3.60742640802207272197799622227, 4.86853994502728291245196181917, 5.57621709765312417170182597516, 6.00862089402620994796524798995, 7.54736716130597860415563334494, 8.315673376621380181281186083999, 9.041205284071703107447268688878, 9.893887134316449520003900315330