L(s) = 1 | + (0.766 − 0.642i)3-s + (−0.173 − 0.984i)5-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 0.342i)29-s + 31-s + (−0.173 − 0.984i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)3-s + (−0.173 − 0.984i)5-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 0.342i)29-s + 31-s + (−0.173 − 0.984i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8998488537 - 1.376248752i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8998488537 - 1.376248752i\) |
\(L(1)\) |
\(\approx\) |
\(1.135408513 - 0.6421203428i\) |
\(L(1)\) |
\(\approx\) |
\(1.135408513 - 0.6421203428i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.173 + 0.984i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.939 + 0.342i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.173 - 0.984i)T \) |
| 53 | \( 1 + (0.173 - 0.984i)T \) |
| 59 | \( 1 + (0.173 + 0.984i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.766 + 0.642i)T \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.939 + 0.342i)T \) |
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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
−23.58437922978990821166355214925, −22.62288282005625323624506675662, −22.16895233035560369865248146976, −21.20256599674044451531630825876, −20.34913289621641993675706392202, −19.52682719308777482041869142578, −18.96257560540285069871547583379, −17.76484538914439575557144826285, −17.06137079165834443223617309402, −15.747438578894610519350051158175, −15.01462098669992216684012188061, −14.72658704663813668838128460925, −13.59436756227365394565201042200, −12.70044178196858604872156224448, −11.45191515072517645308544474939, −10.51713207540524944834429424673, −9.94573260022573624778352767056, −8.930836807287438008920592888940, −7.85990334623712063619862965585, −7.15884131532766356762164354902, −5.950718478175870452269143959376, −4.68153707591875554953770343033, −3.708113802432014216439213008359, −2.89545781895919179757378273130, −1.83054553020180381575459380213,
0.81106517560275154353150962894, 1.86511194649656044692397855649, 3.12195956042945479617452601878, 4.153881898264123683889231774113, 5.23422978599640193919949933330, 6.53366934814170718649605870001, 7.31338510274448500989092692937, 8.54269745150096210711618768206, 8.89144731225034279697663574573, 9.818112804908677134891587683048, 11.47942198975428170675505711101, 11.964718251024002536373683069739, 13.08708683615567071392991180135, 13.66836145932480077909202592637, 14.49348958852804723335698220082, 15.53873776252264423037083728785, 16.51441431330840828902825627452, 17.15943386067697429806609541053, 18.38738760696048800536820846646, 19.11551368899147388847062269173, 19.75728306608086075310874941050, 20.706187133262380608278000812938, 21.18603510385217714437343975462, 22.38864598405438349093028903890, 23.50136448363923365609542184347