L(s) = 1 | + (−0.365 + 0.930i)2-s + (−0.733 − 0.680i)4-s + (−0.826 − 0.563i)5-s + (0.900 − 0.433i)8-s + (0.826 − 0.563i)10-s + (0.988 − 0.149i)11-s + (0.623 + 0.781i)13-s + (0.0747 + 0.997i)16-s + (−0.955 − 0.294i)17-s + (−0.5 − 0.866i)19-s + (0.222 + 0.974i)20-s + (−0.222 + 0.974i)22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.955 + 0.294i)26-s + ⋯ |
L(s) = 1 | + (−0.365 + 0.930i)2-s + (−0.733 − 0.680i)4-s + (−0.826 − 0.563i)5-s + (0.900 − 0.433i)8-s + (0.826 − 0.563i)10-s + (0.988 − 0.149i)11-s + (0.623 + 0.781i)13-s + (0.0747 + 0.997i)16-s + (−0.955 − 0.294i)17-s + (−0.5 − 0.866i)19-s + (0.222 + 0.974i)20-s + (−0.222 + 0.974i)22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.955 + 0.294i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05878677629 + 0.4978262999i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05878677629 + 0.4978262999i\) |
\(L(1)\) |
\(\approx\) |
\(0.5908235393 + 0.2632459326i\) |
\(L(1)\) |
\(\approx\) |
\(0.5908235393 + 0.2632459326i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.365 + 0.930i)T \) |
| 5 | \( 1 + (-0.826 - 0.563i)T \) |
| 11 | \( 1 + (0.988 - 0.149i)T \) |
| 13 | \( 1 + (0.623 + 0.781i)T \) |
| 17 | \( 1 + (-0.955 - 0.294i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.955 + 0.294i)T \) |
| 29 | \( 1 + (0.222 + 0.974i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.733 + 0.680i)T \) |
| 41 | \( 1 + (0.900 - 0.433i)T \) |
| 43 | \( 1 + (-0.900 - 0.433i)T \) |
| 47 | \( 1 + (-0.365 + 0.930i)T \) |
| 53 | \( 1 + (0.733 + 0.680i)T \) |
| 59 | \( 1 + (-0.826 + 0.563i)T \) |
| 61 | \( 1 + (-0.733 + 0.680i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.365 + 0.930i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.623 + 0.781i)T \) |
| 89 | \( 1 + (0.988 + 0.149i)T \) |
| 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
−27.66468144682311158939723128621, −26.71941344783814783618306233514, −25.87920633377894809487066043024, −24.58874943963192474966496791795, −23.05930595623157771190697130883, −22.58021330243362892444148373919, −21.5041468733076323140662005215, −20.22038289762334218166885136142, −19.640701410906589466307975266290, −18.60944961687666605353308725761, −17.74694751969325689078425625330, −16.56881208314179384913517701057, −15.27548152490913922908332377298, −14.14481125843715480774723879835, −12.86406347958250127200812368122, −11.84221214993597411060928545641, −11.0010758800352197262722828046, −10.01691821626512782599823522644, −8.644816968067583924101201848227, −7.77090173327496511965340611419, −6.28988277852378942137353181020, −4.27678530731301281540904640712, −3.51896698087857134217100533972, −1.98879850079978464121588569094, −0.23760451056943978706813090379,
1.34733067230112283382461488552, 3.891243885952827454311480139611, 4.812595637567040127939080234209, 6.353965240346305073463714328183, 7.26750415486846815340750805130, 8.69563832590366999480267572992, 9.082607363756476108340411908690, 10.80318300659946167860429346763, 11.941612148700246621946038468953, 13.33333083079484718115100892608, 14.34356916854443694326828487993, 15.53415619270096233951064188721, 16.22510459622805847768437037710, 17.17059525805302181600072811332, 18.25861440059490049724572648401, 19.44239913848473613957479756493, 20.02641948187355484417275655619, 21.69023831398639001709963103297, 22.746396832595037972804980471680, 23.842242919355458080576274919413, 24.29011666539683688250958731636, 25.46394501021138190413609261179, 26.40754968868966936500296389326, 27.423521530865820878252784236150, 28.00891750156437288812994014361