L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.0581 + 0.998i)3-s + (−0.939 + 0.342i)4-s + (−0.802 − 0.597i)5-s + (0.973 − 0.230i)6-s + (0.396 + 0.918i)7-s + (0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.448 + 0.893i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)12-s + (−0.396 − 0.918i)13-s + (0.835 − 0.549i)14-s + (0.549 − 0.835i)15-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.0581 + 0.998i)3-s + (−0.939 + 0.342i)4-s + (−0.802 − 0.597i)5-s + (0.973 − 0.230i)6-s + (0.396 + 0.918i)7-s + (0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.448 + 0.893i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)12-s + (−0.396 − 0.918i)13-s + (0.835 − 0.549i)14-s + (0.549 − 0.835i)15-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0001358927909 + 0.04992843487i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0001358927909 + 0.04992843487i\) |
\(L(1)\) |
\(\approx\) |
\(0.6782074094 - 0.04556762905i\) |
\(L(1)\) |
\(\approx\) |
\(0.6782074094 - 0.04556762905i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.0581 + 0.998i)T \) |
| 5 | \( 1 + (-0.802 - 0.597i)T \) |
| 7 | \( 1 + (0.396 + 0.918i)T \) |
| 11 | \( 1 + (0.448 + 0.893i)T \) |
| 13 | \( 1 + (-0.396 - 0.918i)T \) |
| 17 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (-0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.998 + 0.0581i)T \) |
| 31 | \( 1 + (0.116 + 0.993i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.642 + 0.766i)T \) |
| 47 | \( 1 + (-0.727 - 0.686i)T \) |
| 53 | \( 1 + (0.116 - 0.993i)T \) |
| 59 | \( 1 + (0.835 + 0.549i)T \) |
| 61 | \( 1 + (0.957 + 0.286i)T \) |
| 67 | \( 1 + (-0.802 + 0.597i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 + (-0.893 + 0.448i)T \) |
| 79 | \( 1 + (-0.396 + 0.918i)T \) |
| 83 | \( 1 + (0.0581 + 0.998i)T \) |
| 89 | \( 1 + (-0.957 - 0.286i)T \) |
| 97 | \( 1 + (-0.802 - 0.597i)T \) |
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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
−18.0598983802941056825899717490, −17.32471922081468538107192091062, −16.84350051617173947979101714674, −16.33252281837283659644478927845, −15.21927748797550620035845028719, −14.64358223740657818969971190950, −14.20995431028759644113113447498, −13.498631366785118904196015749256, −12.9436192212860761956589445612, −11.8833056820043407774775451089, −11.28429609085114484914662251935, −10.68143935169101543287542927933, −9.59340492372475047263168172379, −8.80477228950045689379424662797, −8.04120580193503377964060627688, −7.57377197567604975473585749734, −7.03234066333560404302444557711, −6.357561505736464992568245713685, −5.74073138712555167079932117403, −4.58798211436798878588127776584, −3.89230833906621745901214752297, −3.19843957812310451259007101040, −1.85498162493565005086179583606, −0.989186339706803086726128798797, −0.018266490855355946561449442678,
1.23417668382307018543842521613, 2.30143646022944598492830978949, 2.98322745153966537747107092929, 3.87006014113860280868173299075, 4.40408744972278872979073402328, 5.226753779557982221462823301030, 5.530965909754527892121870374463, 7.14225177180807561239353042310, 8.13169323629728752434088467201, 8.5127464363052081921939310672, 9.18579293674997547266521626359, 9.91028604711591473656781290899, 10.45536178765865710209184483317, 11.3732332005562808344394389542, 11.83679469006213696683753968082, 12.52423100166117125411965351915, 12.89551781129378750826966968572, 14.36178519398873828698983825214, 14.66682461108387899383685419042, 15.30097813760577318291192314643, 16.196924961296706963059781582252, 16.7657989799028925867133908281, 17.51902763078081951019288032171, 18.1354369991861384603506277795, 19.07812058056250483558008132246