L(s) = 1 | − i·5-s + i·7-s − i·11-s + 17-s − i·19-s − 23-s − 25-s − 29-s − i·31-s + 35-s − i·37-s − i·41-s + 43-s − i·47-s − 49-s + ⋯ |
L(s) = 1 | − i·5-s + i·7-s − i·11-s + 17-s − i·19-s − 23-s − 25-s − 29-s − i·31-s + 35-s − i·37-s − i·41-s + 43-s − i·47-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7905489547 - 1.065349498i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7905489547 - 1.065349498i\) |
\(L(1)\) |
\(\approx\) |
\(0.9629445133 - 0.2915561391i\) |
\(L(1)\) |
\(\approx\) |
\(0.9629445133 - 0.2915561391i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 \) |
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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
−27.79564157912691555825504718399, −26.96030373026663149207982039523, −25.99019119690539952334912893419, −25.348264954989370445275522188159, −23.83560862153370405992837407663, −23.02755941793886884233970574127, −22.33274110371869209560765669939, −21.00657954608994555208235565375, −20.122122683751825305711990038561, −19.026510287051759937863753406924, −18.05818060880996978256683002461, −17.11105978464597943396769031542, −15.97722096760541939781661008813, −14.65461410400428289673673393932, −14.12644367150559723128764425694, −12.767811652812311811527990662603, −11.58598707266241941439331337782, −10.34991617817862654786516662247, −9.82220747076534186215505132982, −7.89207187406111397493493820308, −7.16377641453551323797689012228, −5.96278386935602555327514200489, −4.320022370523815341672545757623, −3.21631633378325478307967088815, −1.59910775196490900489612011329,
0.498159412106850487793084297622, 2.157020363890858921805921681585, 3.72932571653588901004922412823, 5.24847680067723015228663918431, 5.97650688912942371433505584158, 7.78389717459136064339750221154, 8.77047962755021680381268702727, 9.59308248953805830577693708282, 11.207364643840820815153053628241, 12.15965601682234077438328471713, 13.08632629856897314912676918400, 14.230082374996387668715940077539, 15.55532578949516553595397888065, 16.31674645082879068141207859943, 17.34830985058547583892854955872, 18.57301916997727361300071338855, 19.42892780387700475110256073061, 20.61471235760106561016425434706, 21.47390024874872394672942764364, 22.316752708775455113140928503232, 23.81788599512921900410089217029, 24.380660703840901600526181669065, 25.334288116201826993633866102645, 26.36003295749242935448144052867, 27.77562987153526294087777853142