L(s) = 1 | + (−0.923 − 0.382i)3-s + (0.382 + 0.923i)5-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.382 − 0.923i)13-s − i·15-s − i·17-s + (−0.382 + 0.923i)19-s + (−0.707 − 0.707i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + (−0.923 − 0.382i)29-s + 31-s + 33-s + (0.382 + 0.923i)37-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)3-s + (0.382 + 0.923i)5-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.382 − 0.923i)13-s − i·15-s − i·17-s + (−0.382 + 0.923i)19-s + (−0.707 − 0.707i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + (−0.923 − 0.382i)29-s + 31-s + 33-s + (0.382 + 0.923i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9762787205 - 0.4617454462i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9762787205 - 0.4617454462i\) |
\(L(1)\) |
\(\approx\) |
\(0.7964576232 + 0.03116165246i\) |
\(L(1)\) |
\(\approx\) |
\(0.7964576232 + 0.03116165246i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.923 - 0.382i)T \) |
| 5 | \( 1 + (0.382 + 0.923i)T \) |
| 11 | \( 1 + (-0.923 + 0.382i)T \) |
| 13 | \( 1 + (0.382 - 0.923i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (-0.382 + 0.923i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
| 29 | \( 1 + (-0.923 - 0.382i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.382 + 0.923i)T \) |
| 41 | \( 1 + (0.707 + 0.707i)T \) |
| 43 | \( 1 + (0.923 - 0.382i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.923 + 0.382i)T \) |
| 59 | \( 1 + (0.382 + 0.923i)T \) |
| 61 | \( 1 + (-0.923 - 0.382i)T \) |
| 67 | \( 1 + (0.923 + 0.382i)T \) |
| 71 | \( 1 + (0.707 - 0.707i)T \) |
| 73 | \( 1 + (-0.707 - 0.707i)T \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 + (-0.382 + 0.923i)T \) |
| 89 | \( 1 + (0.707 - 0.707i)T \) |
| 97 | \( 1 + T \) |
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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
−23.97260415850912975361165467892, −23.236288210468907547190925710337, −22.01073573649003996580405874500, −21.33538520815363141577169002459, −20.87800932119156751704390949800, −19.63865673105599861270177720890, −18.64108634251317308218177212350, −17.58141978214730511253073270080, −17.10755107847391331940714710513, −16.01221761596144768270756190418, −15.716229210110602284047975067194, −14.25918280527982931155434213722, −13.12922280469845549569571496736, −12.58619277130932223604250840036, −11.43446817554881383153450472688, −10.73667522333872797942700343430, −9.65571663309571296434932486301, −8.88816402246085989858361046486, −7.72983538007656943441628052690, −6.34047956726102128450269266458, −5.66323047940893230110367419947, −4.7004813086132855165596880746, −3.85014612999136889075260823135, −2.06956203302998503883018936634, −0.83472156403740780534176003339,
0.42749523033158600934142403832, 1.959854051526102139395979962537, 2.97369049795812060630169901083, 4.4879859918165468763451976692, 5.64163236755641901461568041380, 6.25563933261689527105381531154, 7.36545226186926013263560429349, 8.05602641064244369126261965863, 9.83281887740755894159720905758, 10.405797070362950640135838130933, 11.1985418584272539661716831801, 12.20711998161651051903495830839, 13.113705262490050368319522302795, 13.90125140208977065999489552410, 15.083220180365705844893336669389, 15.88144790808872320056277206030, 16.88495401902137298934881062462, 17.87630285629517001453171574032, 18.34246475152836793730782100709, 18.98481744863340094677600515204, 20.40311066788009270036901465832, 21.18475767896849730954765524404, 22.289625991908932394614335453702, 22.80884240193874998690231915816, 23.38978905481081796824202346055