L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s + i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + i·12-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + i·18-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)22-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s + i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + i·12-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + i·18-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)22-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.781042133 + 2.265329845i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.781042133 + 2.265329845i\) |
\(L(1)\) |
\(\approx\) |
\(1.781429227 + 1.185392844i\) |
\(L(1)\) |
\(\approx\) |
\(1.781429227 + 1.185392844i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.866 - 0.5i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−23.68919326118359130726866675255, −23.067816741707061815064597910067, −21.58600761180417244245161761889, −21.36376317434066249636843791986, −20.27112884439052948519721504990, −19.60897039940174078189612669300, −18.79284124882983243073478318462, −18.144048099626498735151812246655, −16.519424844000536242281547507243, −15.69397899913618396676789588202, −14.50197657906226090665775745395, −14.26306810125189309122234130882, −13.08795606439791376807937793882, −12.61737082329277804801751429627, −11.57005913968123563588681279868, −10.48823734935058639400807929595, −9.62017964714548307576007995157, −8.3937955779006351737663410966, −7.52013053454201026417994089830, −6.29203016635377351284828302489, −5.50734286783121559823672101451, −4.0100747655873168300413848841, −3.31205798566715392232759730027, −2.26260811543794111301954828754, −1.19162626667797002511725839044,
2.143008234248460055626422373912, 2.90872840840612054689580274236, 4.12821071083290114099428961143, 4.77257748509098395433228172161, 5.90878561386029188354063196008, 7.2528634767517908918150324160, 7.83713766092552360817299860429, 8.91150510034420118962717930804, 9.9622037383572165327382018845, 10.98242500696923627699575214007, 12.235557990601397869969072656706, 13.011484549515517177266416373707, 13.94278519974917100951129876694, 14.613613832938397194701002529627, 15.462391923408414086088285514, 16.04354867708993677406944109700, 17.04607236627829617177198986243, 18.10202861560843141231129878718, 19.292948791013750936171227426969, 20.35088011042320849739994408818, 20.77072839832779533794233252696, 21.70685366970613511461315220888, 22.42263281793434528820245853964, 23.39384438807330248587885887172, 24.18365764765016023036662739684