L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)8-s + (−0.955 − 0.294i)11-s + (0.955 + 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 + 0.997i)22-s + (0.826 + 0.563i)23-s + (0.0747 − 0.997i)26-s + (−0.0747 − 0.997i)29-s − 31-s + (−0.900 − 0.433i)32-s + (−0.955 + 0.294i)34-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)8-s + (−0.955 − 0.294i)11-s + (0.955 + 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 + 0.997i)22-s + (0.826 + 0.563i)23-s + (0.0747 − 0.997i)26-s + (−0.0747 − 0.997i)29-s − 31-s + (−0.900 − 0.433i)32-s + (−0.955 + 0.294i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4236452848 - 0.9757427619i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4236452848 - 0.9757427619i\) |
\(L(1)\) |
\(\approx\) |
\(0.7264138550 - 0.4458983851i\) |
\(L(1)\) |
\(\approx\) |
\(0.7264138550 - 0.4458983851i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 11 | \( 1 + (-0.955 - 0.294i)T \) |
| 13 | \( 1 + (0.955 + 0.294i)T \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.826 + 0.563i)T \) |
| 29 | \( 1 + (-0.0747 - 0.997i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.826 + 0.563i)T \) |
| 41 | \( 1 + (-0.988 + 0.149i)T \) |
| 43 | \( 1 + (0.988 + 0.149i)T \) |
| 47 | \( 1 + (0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.826 + 0.563i)T \) |
| 59 | \( 1 + (0.623 - 0.781i)T \) |
| 61 | \( 1 + (0.900 + 0.433i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.900 - 0.433i)T \) |
| 73 | \( 1 + (0.955 - 0.294i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.955 + 0.294i)T \) |
| 89 | \( 1 + (-0.733 - 0.680i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.92376528668131423555903835030, −18.97421643142359999155469772920, −18.3827264699359115230007357192, −17.90936930409250774824863356873, −17.01419780200951397895160197223, −16.34217820070383392853681710008, −15.7396011463245653478780249409, −15.00659354089239260719238960333, −14.426574831589910423283516578839, −13.48307563228109221183874811158, −12.938340678210593373278736398428, −12.2282743685671315830576734343, −10.70478605112465257363864094548, −10.59045598144358109059624306946, −9.53576275637093655020697136437, −8.60964423401639818363619171160, −8.2127199214292093100288964584, −7.2591223265336475911519774259, −6.655325419295710249324298477513, −5.47366563671457669057218242748, −5.380877957451158770324827774177, −4.047449472458793169238154969262, −3.4277878575850730409110801367, −2.012736060261931043781385177475, −0.98701270010624627310333046412,
0.475192896677169112213663438130, 1.466224672476565396918816280643, 2.55183151849333152838095008107, 3.15342166085568956818138969495, 4.05832825180811007707723623584, 5.02252247715038879749489335644, 5.59724300295110669976545604806, 6.91339863113218105120375626601, 7.68972602023639313348355165762, 8.541595803734065900243972132783, 9.22377870587821262148927011315, 9.874854490568285433260634133766, 10.924443824528206001470606945798, 11.20191339769311043382630493046, 12.01971055393428319867621700195, 12.96475560492212423459783454186, 13.56336832555100110728125640856, 13.95644049885804461243270860596, 15.22314933937274669944482244908, 15.901995903341124849160825652982, 16.68228714744607253823552289386, 17.58283716899700910602823898616, 18.192050188731756218537255370445, 18.77658260115463446075400807116, 19.40249081279678629665263461813