L(s) = 1 | + (0.221 − 1.39i)2-s + (−1.90 − 0.618i)4-s + (1.17 + 1.90i)5-s + (1.90 + 1.90i)7-s + (−1.28 + 2.52i)8-s + (2.91 − 1.22i)10-s + 3.23·11-s + (−0.726 + 0.726i)13-s + (3.07 − 2.23i)14-s + (3.23 + 2.35i)16-s + (1 − i)17-s + 2i·19-s + (−1.06 − 4.34i)20-s + (0.715 − 4.52i)22-s + (4.25 − 4.25i)23-s + ⋯ |
L(s) = 1 | + (0.156 − 0.987i)2-s + (−0.951 − 0.309i)4-s + (0.525 + 0.850i)5-s + (0.718 + 0.718i)7-s + (−0.453 + 0.891i)8-s + (0.922 − 0.386i)10-s + 0.975·11-s + (−0.201 + 0.201i)13-s + (0.822 − 0.597i)14-s + (0.809 + 0.587i)16-s + (0.242 − 0.242i)17-s + 0.458i·19-s + (−0.237 − 0.971i)20-s + (0.152 − 0.963i)22-s + (0.886 − 0.886i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50950 - 0.362399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50950 - 0.362399i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.221 + 1.39i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.17 - 1.90i)T \) |
good | 7 | \( 1 + (-1.90 - 1.90i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + (0.726 - 0.726i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - 17iT^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + (-4.25 + 4.25i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.15T + 29T^{2} \) |
| 31 | \( 1 + 8.50iT - 31T^{2} \) |
| 37 | \( 1 + (-0.726 - 0.726i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 + (-4.61 - 4.61i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.35 - 3.35i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.07 - 3.07i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.472iT - 59T^{2} \) |
| 61 | \( 1 - 0.898iT - 61T^{2} \) |
| 67 | \( 1 + (4.61 - 4.61i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 + (4.70 + 4.70i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.90T + 79T^{2} \) |
| 83 | \( 1 + (-6.61 - 6.61i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.47iT - 89T^{2} \) |
| 97 | \( 1 + (-4.23 + 4.23i)T - 97iT^{2} \) |
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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
−11.39744288001080489829827430573, −10.64743120959013420927416213995, −9.576156577589139111376373050695, −8.993373406076395543879676608144, −7.74321178630606147372369434094, −6.34033119954016537375803470742, −5.39038999632128707983450944441, −4.14539672535001973180135470108, −2.82370774028713957140414369073, −1.72790732431045930482965840515,
1.26476519738723477677176822607, 3.72688406204262459653840886439, 4.79858600404903128705383254023, 5.56695520162272704610009710773, 6.80390176474097542758067662395, 7.64789109178907228429352056954, 8.730845019442835716695226901161, 9.314705649198891177649218340985, 10.41063531567298730401458357649, 11.66861078478206941731444922161