L(s) = 1 | + (−0.382 + 0.923i)5-s + (−0.707 − 0.707i)7-s + (0.923 + 0.382i)11-s + (0.382 + 0.923i)13-s − i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (−0.923 + 0.382i)29-s + 31-s + (0.923 − 0.382i)35-s + (−0.382 + 0.923i)37-s + (0.707 − 0.707i)41-s + (0.923 + 0.382i)43-s − i·47-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)5-s + (−0.707 − 0.707i)7-s + (0.923 + 0.382i)11-s + (0.382 + 0.923i)13-s − i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (−0.923 + 0.382i)29-s + 31-s + (0.923 − 0.382i)35-s + (−0.382 + 0.923i)37-s + (0.707 − 0.707i)41-s + (0.923 + 0.382i)43-s − i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7704197077 + 0.5713821971i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7704197077 + 0.5713821971i\) |
\(L(1)\) |
\(\approx\) |
\(0.9089399805 + 0.2561186283i\) |
\(L(1)\) |
\(\approx\) |
\(0.9089399805 + 0.2561186283i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.382 + 0.923i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.923 + 0.382i)T \) |
| 13 | \( 1 + (0.382 + 0.923i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (0.382 + 0.923i)T \) |
| 23 | \( 1 + (-0.707 + 0.707i)T \) |
| 29 | \( 1 + (-0.923 + 0.382i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.382 + 0.923i)T \) |
| 41 | \( 1 + (0.707 - 0.707i)T \) |
| 43 | \( 1 + (0.923 + 0.382i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.923 - 0.382i)T \) |
| 59 | \( 1 + (0.382 - 0.923i)T \) |
| 61 | \( 1 + (-0.923 + 0.382i)T \) |
| 67 | \( 1 + (0.923 - 0.382i)T \) |
| 71 | \( 1 + (0.707 + 0.707i)T \) |
| 73 | \( 1 + (0.707 - 0.707i)T \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 + (-0.382 - 0.923i)T \) |
| 89 | \( 1 + (0.707 + 0.707i)T \) |
| 97 | \( 1 - T \) |
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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
−27.00951079827215540125396640835, −25.81384699065526517650376604885, −24.7572488318086396519544585642, −24.36777935335100117657658346668, −22.87001454899104009767308830534, −22.313397098136815353652806120866, −21.05487166556166709953182512503, −20.08374347230971349989176110974, −19.36308905979048812678358370433, −18.231176997879133574845781465291, −17.10504787195451735737319207164, −16.028554510433726810835083823483, −15.52434171077037933544073228847, −14.05031383497374315548809419838, −12.954534644516846591815433736468, −12.144863905279342745022591629040, −11.18752872217067576958413728255, −9.56271699032335374240612173296, −8.92059585512714649082671863072, −7.78131004520313083672210242495, −6.33314429177044550565798428114, −5.304488657080333626699896844027, −3.99643315049191803411315277472, −2.7069270335384313977982506703, −0.7979712642686732686142777796,
1.71255706386065786021717157513, 3.49416128349911271833936101621, 4.08867786770784993639341791308, 6.11854547280180245031037520969, 6.85191340942384255611768321513, 7.921555537800598264741259744618, 9.42826908151735781654801196630, 10.323270343464711238677438699064, 11.39385659265008978868777335654, 12.371394941533524688280627123011, 13.75706796901486980976794862909, 14.45906300997291761687385583381, 15.59168946143793403980894568085, 16.61059638851498803761584834493, 17.558293195285664745770588613582, 18.86170989759774937036690325819, 19.41970088682467717272376227420, 20.43490182781641802085735656260, 21.74946445735226705836623405989, 22.59445282479406489577297575224, 23.29961988476842615240159737661, 24.29453424590591066322567919953, 25.74794643785004066163396397515, 26.147411713085298299332654022316, 27.16792707369892515317074600173