L(s) = 1 | + (−0.134 − 0.990i)2-s + (−0.753 + 0.657i)3-s + (−0.963 + 0.266i)4-s + (0.753 + 0.657i)6-s + (0.393 + 0.919i)8-s + (0.134 − 0.990i)9-s + (0.134 + 0.990i)11-s + (0.550 − 0.834i)12-s + (0.691 + 0.722i)13-s + (0.858 − 0.512i)16-s + (0.963 + 0.266i)17-s − 18-s + (−0.809 + 0.587i)19-s + (0.963 − 0.266i)22-s + (0.550 + 0.834i)23-s + (−0.900 − 0.433i)24-s + ⋯ |
L(s) = 1 | + (−0.134 − 0.990i)2-s + (−0.753 + 0.657i)3-s + (−0.963 + 0.266i)4-s + (0.753 + 0.657i)6-s + (0.393 + 0.919i)8-s + (0.134 − 0.990i)9-s + (0.134 + 0.990i)11-s + (0.550 − 0.834i)12-s + (0.691 + 0.722i)13-s + (0.858 − 0.512i)16-s + (0.963 + 0.266i)17-s − 18-s + (−0.809 + 0.587i)19-s + (0.963 − 0.266i)22-s + (0.550 + 0.834i)23-s + (−0.900 − 0.433i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6814708202 + 0.4300269727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6814708202 + 0.4300269727i\) |
\(L(1)\) |
\(\approx\) |
\(0.7264528543 + 0.01894287063i\) |
\(L(1)\) |
\(\approx\) |
\(0.7264528543 + 0.01894287063i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.134 - 0.990i)T \) |
| 3 | \( 1 + (-0.753 + 0.657i)T \) |
| 11 | \( 1 + (0.134 + 0.990i)T \) |
| 13 | \( 1 + (0.691 + 0.722i)T \) |
| 17 | \( 1 + (0.963 + 0.266i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.550 + 0.834i)T \) |
| 29 | \( 1 + (-0.0448 - 0.998i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 37 | \( 1 + (0.550 - 0.834i)T \) |
| 41 | \( 1 + (-0.995 + 0.0896i)T \) |
| 43 | \( 1 + (0.222 + 0.974i)T \) |
| 47 | \( 1 + (-0.983 - 0.178i)T \) |
| 53 | \( 1 + (0.963 - 0.266i)T \) |
| 59 | \( 1 + (0.858 - 0.512i)T \) |
| 61 | \( 1 + (0.936 - 0.351i)T \) |
| 67 | \( 1 + (0.809 - 0.587i)T \) |
| 71 | \( 1 + (-0.963 + 0.266i)T \) |
| 73 | \( 1 + (0.691 - 0.722i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.983 + 0.178i)T \) |
| 89 | \( 1 + (-0.691 + 0.722i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−21.23648851089659692625107429820, −20.02626809209683063177509098661, −18.98195793074176444297250724143, −18.593834018086064282415047203181, −17.9248515201555120355479397028, −16.9232164444590308957812879309, −16.6277859262509513340669273774, −15.80784234764857820246701496311, −14.86000802403213716008044930215, −14.05179515238369053493456980506, −13.15759725282593194072016883966, −12.794492799151620864232606578092, −11.59319182597241695046654692521, −10.7708334530563563418405906339, −10.0395878409432768930482474947, −8.668589410394648374923558026040, −8.31796346397446771945349118999, −7.22632142032498880907613299048, −6.62075584681960553756182209127, −5.70422382080919886638912285743, −5.28581269513550625427244071611, −4.11120020947866139626263422041, −2.95333386438666005408777443894, −1.3329915134240939306309473626, −0.46358057233166255275816151619,
1.16715286026852640204945959502, 2.05042150326510236827374597197, 3.49969705866120927215357946373, 4.01108935858827270646923089388, 4.91237389936666393553256094183, 5.73603506660844677668586051635, 6.786805544655022519822884313444, 7.953099122055452869299098662136, 8.97634357422927596634468243119, 9.72081894410096019707943492436, 10.25712017635297663916598660979, 11.17832486883774524051957869403, 11.720269288049139411204510559813, 12.52869016059761299356557997895, 13.18279612932841581155113801461, 14.42099196545410358746655184538, 14.93670849543169051577638319324, 16.10638478115048320362399326073, 16.84142277650510140823169912021, 17.485623559905017724157476993484, 18.22841955830880627317977641322, 18.98432682919988224496493140619, 19.81584860447309950112405350766, 20.76553078803086495151407166997, 21.235373491254228752716147666671