L(s) = 1 | + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)13-s − i·17-s − 19-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s − i·53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (−0.866 − 0.5i)67-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)13-s − i·17-s − 19-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s − i·53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (−0.866 − 0.5i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.237635397 + 0.1991513553i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.237635397 + 0.1991513553i\) |
\(L(1)\) |
\(\approx\) |
\(0.8900170081 - 0.06135771474i\) |
\(L(1)\) |
\(\approx\) |
\(0.8900170081 - 0.06135771474i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.866 + 0.5i)T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 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 | \( 1 - T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.866 + 0.5i)T \) |
| 89 | \( 1 - T \) |
| 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
−19.34807609857657746223592241107, −18.587185779888321781989836185737, −17.618252740120050662123163282743, −17.2216619499667685587662158438, −16.47761983782241994373889704320, −15.52802269174803084213483212795, −14.93985888334863950247377994047, −14.40123776025045732210794926665, −13.387914270635767069321564793764, −12.68658133545510005863582146386, −12.18329531227107125431624136102, −11.28269645484014339489067486632, −10.28956010448854710397947750749, −10.05287656034609874288597015990, −8.88083649093009118162449949562, −8.35634821834252927306662265694, −7.33005416599871781540910410913, −6.78680541622839239184246940373, −5.905797863412553624584097829655, −4.85113486962745776387460120388, −4.42138752872923838304453778073, −3.34807296227260339484945767131, −2.301075867638482915116983920528, −1.75131056907823179868041872343, −0.341146106782012182360044265684,
0.52720228585713626987689151661, 1.57596102129342307137616274119, 2.913986593388190513272913687235, 3.04676989410381683242417601745, 4.542915435151040133788479830336, 5.01184831085659428592133861667, 5.96245465919127306768931025724, 6.73620138995909088526326826456, 7.62112581694668440912333026370, 8.25945442045354338829196490463, 9.09506478955269357303689133492, 9.88274504046589048387821249613, 10.61659008712985518568766705339, 11.35337398631543734621289920697, 12.05162929156105705057049846621, 12.976647156910950346040459849269, 13.46190993711692061176729797770, 14.30600018263728462913544946943, 15.11868973268971620861973585798, 15.61498437716260192878633939125, 16.693075315097598906268730591874, 16.93320396914442715088268515319, 18.01769710234873277348558565742, 18.52323991208454730090657470037, 19.358573063945502054558831428115