L(s) = 1 | + (−0.555 + 0.831i)5-s + (−0.923 + 0.382i)7-s + (0.195 + 0.980i)11-s + (−0.555 − 0.831i)13-s + (0.707 + 0.707i)17-s + (−0.831 + 0.555i)19-s + (−0.382 + 0.923i)23-s + (−0.382 − 0.923i)25-s + (−0.195 + 0.980i)29-s − i·31-s + (0.195 − 0.980i)35-s + (0.831 + 0.555i)37-s + (0.382 − 0.923i)41-s + (−0.980 + 0.195i)43-s + (−0.707 − 0.707i)47-s + ⋯ |
L(s) = 1 | + (−0.555 + 0.831i)5-s + (−0.923 + 0.382i)7-s + (0.195 + 0.980i)11-s + (−0.555 − 0.831i)13-s + (0.707 + 0.707i)17-s + (−0.831 + 0.555i)19-s + (−0.382 + 0.923i)23-s + (−0.382 − 0.923i)25-s + (−0.195 + 0.980i)29-s − i·31-s + (0.195 − 0.980i)35-s + (0.831 + 0.555i)37-s + (0.382 − 0.923i)41-s + (−0.980 + 0.195i)43-s + (−0.707 − 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07862628854 - 0.1007502778i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07862628854 - 0.1007502778i\) |
\(L(1)\) |
\(\approx\) |
\(0.6940245302 + 0.1827811781i\) |
\(L(1)\) |
\(\approx\) |
\(0.6940245302 + 0.1827811781i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.555 + 0.831i)T \) |
| 7 | \( 1 + (-0.923 + 0.382i)T \) |
| 11 | \( 1 + (0.195 + 0.980i)T \) |
| 13 | \( 1 + (-0.555 - 0.831i)T \) |
| 17 | \( 1 + (0.707 + 0.707i)T \) |
| 19 | \( 1 + (-0.831 + 0.555i)T \) |
| 23 | \( 1 + (-0.382 + 0.923i)T \) |
| 29 | \( 1 + (-0.195 + 0.980i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.831 + 0.555i)T \) |
| 41 | \( 1 + (0.382 - 0.923i)T \) |
| 43 | \( 1 + (-0.980 + 0.195i)T \) |
| 47 | \( 1 + (-0.707 - 0.707i)T \) |
| 53 | \( 1 + (-0.195 - 0.980i)T \) |
| 59 | \( 1 + (0.555 - 0.831i)T \) |
| 61 | \( 1 + (0.980 + 0.195i)T \) |
| 67 | \( 1 + (0.980 + 0.195i)T \) |
| 71 | \( 1 + (-0.923 + 0.382i)T \) |
| 73 | \( 1 + (-0.923 - 0.382i)T \) |
| 79 | \( 1 + (0.707 - 0.707i)T \) |
| 83 | \( 1 + (-0.831 + 0.555i)T \) |
| 89 | \( 1 + (-0.382 - 0.923i)T \) |
| 97 | \( 1 - iT \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.51031857702268004363369660516, −23.64663645838024539456928188508, −22.99290331783853589425157056324, −21.87775205342188764700982409988, −21.090540299443085240652647813990, −20.01731953961317513584479572248, −19.37458494465386281546283945377, −18.70293259352325625540617626282, −17.22705446905042477785396041670, −16.39320511460302413010174582442, −16.11680177765794154039199269442, −14.75707643532375195231463750420, −13.74799461194719644475942469674, −12.88267265236453393598018084063, −12.02710891878187866550835555396, −11.14585827129637117452702025579, −9.87834012676972402901627702931, −9.06190324082268470311363826586, −8.12828503335331315358553723630, −7.00354575701904398472157391944, −6.037770706273497695341562959718, −4.730675271205801042218047816898, −3.86822363603317665276877147115, −2.68571826214489967725434254772, −0.9535189788979928598924825137,
0.04248444739311075056261525052, 1.9996490175961480654634875411, 3.17036071840640156542545654410, 3.99646668634753653379700891423, 5.48847314900607054165334019148, 6.5036906092018088582421305601, 7.403538740016460790440439896849, 8.31921285022808510654507210537, 9.814668316462546033644168531777, 10.19897711963733108859903493112, 11.5004754104154078069182542554, 12.41985829989411598803044961509, 13.07998318117400439243364639048, 14.65284731475236040304507851765, 14.99550178328908729647488870520, 15.951418295387993125582559318092, 17.01362761375406531455528780045, 17.97924545517752800804988216369, 18.92134356480105521723752231357, 19.55609430257398857632128324497, 20.36954411111737408472612495593, 21.710519267478123782588118548935, 22.368321380825607864345243908620, 23.08702035817385466277995359061, 23.8169545891042064010923485680