L(s) = 1 | + (0.464 + 0.885i)2-s + (0.935 − 0.354i)3-s + (−0.568 + 0.822i)4-s + (0.663 + 0.748i)5-s + (0.748 + 0.663i)6-s + (−0.885 + 0.464i)7-s + (−0.992 − 0.120i)8-s + (0.748 − 0.663i)9-s + (−0.354 + 0.935i)10-s + (0.970 + 0.239i)11-s + (−0.239 + 0.970i)12-s + (0.568 + 0.822i)13-s + (−0.822 − 0.568i)14-s + (0.885 + 0.464i)15-s + (−0.354 − 0.935i)16-s + (−0.120 − 0.992i)17-s + ⋯ |
L(s) = 1 | + (0.464 + 0.885i)2-s + (0.935 − 0.354i)3-s + (−0.568 + 0.822i)4-s + (0.663 + 0.748i)5-s + (0.748 + 0.663i)6-s + (−0.885 + 0.464i)7-s + (−0.992 − 0.120i)8-s + (0.748 − 0.663i)9-s + (−0.354 + 0.935i)10-s + (0.970 + 0.239i)11-s + (−0.239 + 0.970i)12-s + (0.568 + 0.822i)13-s + (−0.822 − 0.568i)14-s + (0.885 + 0.464i)15-s + (−0.354 − 0.935i)16-s + (−0.120 − 0.992i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0724 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0724 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.726288783 + 1.856159017i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.726288783 + 1.856159017i\) |
\(L(1)\) |
\(\approx\) |
\(1.493400280 + 0.9489759423i\) |
\(L(1)\) |
\(\approx\) |
\(1.493400280 + 0.9489759423i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (0.464 + 0.885i)T \) |
| 3 | \( 1 + (0.935 - 0.354i)T \) |
| 5 | \( 1 + (0.663 + 0.748i)T \) |
| 7 | \( 1 + (-0.885 + 0.464i)T \) |
| 11 | \( 1 + (0.970 + 0.239i)T \) |
| 13 | \( 1 + (0.568 + 0.822i)T \) |
| 17 | \( 1 + (-0.120 - 0.992i)T \) |
| 19 | \( 1 + (-0.822 + 0.568i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (0.970 - 0.239i)T \) |
| 31 | \( 1 + (-0.239 - 0.970i)T \) |
| 37 | \( 1 + (0.354 + 0.935i)T \) |
| 41 | \( 1 + (0.239 - 0.970i)T \) |
| 43 | \( 1 + (0.354 - 0.935i)T \) |
| 47 | \( 1 + (-0.748 - 0.663i)T \) |
| 59 | \( 1 + (0.748 + 0.663i)T \) |
| 61 | \( 1 + (-0.992 - 0.120i)T \) |
| 67 | \( 1 + (-0.822 - 0.568i)T \) |
| 71 | \( 1 + (0.935 + 0.354i)T \) |
| 73 | \( 1 + (-0.992 + 0.120i)T \) |
| 79 | \( 1 + (-0.464 + 0.885i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (0.120 + 0.992i)T \) |
| 97 | \( 1 + (-0.748 + 0.663i)T \) |
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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
−32.6490751152542209290505349114, −31.84312893736409712608379351373, −30.39790587787840357972573220348, −29.63945805907900677936059092363, −28.30606487866728160465978750574, −27.33924670845019970625960477557, −25.86074568344269835170021739111, −24.84537670859846340992515569811, −23.41627403023693800856003937344, −21.930773977439727144789975488893, −21.19373612219745404347375751596, −19.83240414466390454287029620593, −19.53436358723526289271678850994, −17.60856262700646689100014601763, −16.00074623111160853949403028358, −14.53746391632666482999721006218, −13.373457038744031968700941487662, −12.732462665718251209976687945534, −10.672249994926160758440049823402, −9.58310705839715911657038751150, −8.635002483010928965276911935308, −6.1495441154390403486361295844, −4.37353199011836464192324616797, −3.167737202279701417632959283207, −1.37631575552000512064565760512,
2.526224966309040285529292462406, 3.94678409033780381558273231369, 6.272525271920534424343677266108, 6.9169967249092725091726617739, 8.71338214824020373558435316541, 9.67059743379113481887876134533, 12.13750137941733710301356415448, 13.45418738661702868277311613246, 14.27843457872534471058697408331, 15.28540708746868814798421248040, 16.67954736733792963670369317421, 18.23789034141238472044601356106, 19.05929099883331793397498014669, 20.84574434594820593251714107176, 22.02085287262140674078992123873, 22.99357704633409183759670653951, 24.55536186097192426766533175626, 25.449297643691450338457672020070, 25.98504672998857851179212766578, 27.17772799657131764009565820226, 29.21538775289216418416493372352, 30.32120349292735044122003236776, 31.221837200084834631448568958349, 32.32951983849496554673589351489, 33.165276212767087551486412717340