L(s) = 1 | − i·2-s + 2.62i·3-s − 4-s + (1.70 − 1.44i)5-s + 2.62·6-s + 1.83i·7-s + i·8-s − 3.89·9-s + (−1.44 − 1.70i)10-s + 4.19·11-s − 2.62i·12-s − 0.369i·13-s + 1.83·14-s + (3.79 + 4.47i)15-s + 16-s + 5.08i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.51i·3-s − 0.5·4-s + (0.762 − 0.646i)5-s + 1.07·6-s + 0.692i·7-s + 0.353i·8-s − 1.29·9-s + (−0.457 − 0.539i)10-s + 1.26·11-s − 0.757i·12-s − 0.102i·13-s + 0.489·14-s + (0.980 + 1.15i)15-s + 0.250·16-s + 1.23i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37764 + 0.505616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37764 + 0.505616i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (-1.70 + 1.44i)T \) |
| 37 | \( 1 + iT \) |
good | 3 | \( 1 - 2.62iT - 3T^{2} \) |
| 7 | \( 1 - 1.83iT - 7T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 + 0.369iT - 13T^{2} \) |
| 17 | \( 1 - 5.08iT - 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 - 5.62iT - 23T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 41 | \( 1 + 8.01T + 41T^{2} \) |
| 43 | \( 1 + 2.27iT - 43T^{2} \) |
| 47 | \( 1 + 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 9.94iT - 53T^{2} \) |
| 59 | \( 1 - 5.34T + 59T^{2} \) |
| 61 | \( 1 + 9.79T + 61T^{2} \) |
| 67 | \( 1 + 1.85iT - 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 + 8.09iT - 73T^{2} \) |
| 79 | \( 1 + 6.06T + 79T^{2} \) |
| 83 | \( 1 + 8.93iT - 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 6.05iT - 97T^{2} \) |
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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
−11.45242072300559178676905444079, −10.27921196098941964021646622906, −9.878054097338939385932952051973, −8.892122956925646261534946046606, −8.550835254716635621981619987209, −6.29396687202998277697633103484, −5.36560315906133467785132791607, −4.39486599754975174503097917282, −3.49608328756568861810200040506, −1.84213289369450174461901842888,
1.15742013890095378258044901024, 2.68946954117892575433683834986, 4.45030878689047533762249394042, 6.06058979012472761341266422193, 6.67693257951949566975457504481, 7.15177947766005251355371343765, 8.238062434702185586076647911974, 9.269728557592769380594043836003, 10.30885637107653620888659353485, 11.47529982634392767198273018490