L(s) = 1 | + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (0.809 + 0.587i)33-s + (−0.309 + 0.951i)37-s + (0.309 + 0.951i)39-s + (−0.309 + 0.951i)41-s − 43-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (0.809 + 0.587i)33-s + (−0.309 + 0.951i)37-s + (0.309 + 0.951i)39-s + (−0.309 + 0.951i)41-s − 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.275086520 - 0.5048424672i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.275086520 - 0.5048424672i\) |
\(L(1)\) |
\(\approx\) |
\(0.8482965079 + 0.01580971261i\) |
\(L(1)\) |
\(\approx\) |
\(0.8482965079 + 0.01580971261i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (-0.309 - 0.951i)T \) |
| 13 | \( 1 + (0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 + (0.309 + 0.951i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.809 + 0.587i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.809 - 0.587i)T \) |
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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
−20.83739128196483789468285290160, −19.87142875607720735659419484278, −18.81148327174384433299007074891, −18.62737079543812928102240924179, −17.65677080324527367058159981016, −16.95356988977657363956643021270, −16.3417139236514072199592168215, −15.53147366151678283784589913514, −14.47203463964820054465850574792, −13.82137058304016804836512316084, −12.82740532573140647118868922749, −12.24970726660557811967937561186, −11.65488117537749413360312337711, −10.60608124938922751416294877589, −10.11583042023605166322563606304, −8.97931250018835605528936862166, −8.0487768360960003018120425767, −7.15224203131632059200748778260, −6.60023562622019449379902303252, −5.67409142354327936133672268827, −4.79649220822213687781868667644, −4.06851293941470991113972894281, −2.588742457121206843780816295576, −1.78462462577865785598603145115, −0.7528930433796518360264976760,
0.43106323093623969530565032050, 1.28280340115032555070479579241, 2.92076459921437032651425507248, 3.5739792989677011073671710985, 4.64846511045669928048008699256, 5.471880683167241979603082353338, 6.080758337570389175257212533661, 6.95103524558104490630237950637, 8.19594878882604043863485604934, 8.71416972595668829828433314865, 10.1105198899457127048620981736, 10.261316445294965394531648836762, 11.31814877025608362744679427530, 11.834147812421055053577059338244, 12.8979714159611228092518316682, 13.4729679801199061266827525690, 14.62350867912804227788964091495, 15.43002560050535432607074400270, 15.873322003907805163276487664, 16.877663375302043221904553580847, 17.317934572936901620304650065102, 18.147945503169270934429771935, 18.96721242852042322156063967582, 19.742621611232344515571663116338, 20.755782956525558038838528539655