| L(s) = 1 | − i·2-s + i·3-s + 4-s + 6-s + 2i·7-s − 3i·8-s − 9-s + 4·11-s + i·12-s − 6i·13-s + 2·14-s − 16-s − 4i·17-s + i·18-s − 2·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s + 0.5·4-s + 0.408·6-s + 0.755i·7-s − 1.06i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s − 1.66i·13-s + 0.534·14-s − 0.250·16-s − 0.970i·17-s + 0.235i·18-s − 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.134971859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.134971859\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.417454879385253373474424084177, −8.589040329131168966166106550331, −7.68721855317698650630287436133, −6.65457617594089690359713068519, −5.93299524467769070122228550734, −5.02727829589196191750834264811, −3.90690624192849382584039561356, −3.07739387530587078474388351717, −2.31182557291014674616625267859, −0.858866595082527605210475615196,
1.37462194986124042147049610851, 2.20231908504507520786162751786, 3.72392062771303187982747713229, 4.47095842857512516853988855118, 5.80167725380005007568105735389, 6.51388362499070437883134406050, 6.92881675065301117980941161416, 7.63292403833841214754521716499, 8.579325015735999938114211440236, 9.150077253812395240023615614758
