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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6520215181 - 0.07368321742i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6520215181 - 0.07368321742i\) |
\(L(1)\) |
\(\approx\) |
\(0.5616792506 - 0.1337864386i\) |
\(L(1)\) |
\(\approx\) |
\(0.5616792506 - 0.1337864386i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−33.21033919235834912210207186122, −32.51733017242163942417733347521, −30.732690730442239783999524942028, −29.61899536161661579903965319127, −28.34414456155551977549656150713, −27.3346466799053395884772586749, −26.366577831452915428852861257521, −25.068562670967316639857521143114, −23.86471599900087238612202211961, −22.773662708426880937653882247186, −22.39247553119063226257846503581, −19.719799261559173669966714358536, −18.85609878790610369991124434629, −17.74189024223082961955061900653, −16.56759759432066821041029934414, −15.68716350996335423344670479012, −14.21988611405289132978131104985, −12.71360161385347455822517850905, −11.04139414699101542300972152580, −10.029648352030280323598427784092, −8.05985051822510659142886274130, −6.781804532812151577860782509608, −6.01472429153184783679318541306, −3.97104035173723032165264670571, −0.70348090182065996456245498293,
0.99556171507493885097412665946, 3.57310421837356414602630288779, 4.87665022653219787225984091157, 6.8863799965698421056089660932, 8.93322735665775311777423070736, 9.75565855673807396025602286463, 11.61773906104938398744835432012, 11.93217260050464151694703400533, 13.37223934259121379014288757267, 15.81576096753828542963953671610, 16.54477872593990623225208791310, 17.81121938741046804410668036062, 19.11377607204480975720562022097, 20.15812724219843410397963382648, 21.47326440306048724280080607217, 22.466112294975764040193574027125, 23.45434361877314586081042800659, 25.13809874315550571974268085954, 26.71857324314472984500184666599, 27.64408509698383129879425741715, 28.53795497417586848549755560256, 29.16245865348261517181041213228, 30.64848305321330307529042223381, 31.82914700262738242036535887014, 32.86758732305956094774794649439