L(s) = 1 | + (−0.382 − 0.923i)3-s + (0.980 + 0.195i)5-s + (−0.980 + 0.195i)7-s + (−0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (0.195 − 0.980i)13-s + (−0.195 − 0.980i)15-s + (0.195 − 0.980i)17-s + (0.980 + 0.195i)19-s + (0.555 + 0.831i)21-s + (0.831 − 0.555i)23-s + (0.923 + 0.382i)25-s + (0.923 + 0.382i)27-s + (−0.831 + 0.555i)29-s + (0.923 − 0.382i)31-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)3-s + (0.980 + 0.195i)5-s + (−0.980 + 0.195i)7-s + (−0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (0.195 − 0.980i)13-s + (−0.195 − 0.980i)15-s + (0.195 − 0.980i)17-s + (0.980 + 0.195i)19-s + (0.555 + 0.831i)21-s + (0.831 − 0.555i)23-s + (0.923 + 0.382i)25-s + (0.923 + 0.382i)27-s + (−0.831 + 0.555i)29-s + (0.923 − 0.382i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.126205929 - 0.5428925082i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.126205929 - 0.5428925082i\) |
\(L(1)\) |
\(\approx\) |
\(1.018130164 - 0.2737436623i\) |
\(L(1)\) |
\(\approx\) |
\(1.018130164 - 0.2737436623i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 97 | \( 1 \) |
good | 3 | \( 1 + (-0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.980 + 0.195i)T \) |
| 7 | \( 1 + (-0.980 + 0.195i)T \) |
| 11 | \( 1 + (0.382 + 0.923i)T \) |
| 13 | \( 1 + (0.195 - 0.980i)T \) |
| 17 | \( 1 + (0.195 - 0.980i)T \) |
| 19 | \( 1 + (0.980 + 0.195i)T \) |
| 23 | \( 1 + (0.831 - 0.555i)T \) |
| 29 | \( 1 + (-0.831 + 0.555i)T \) |
| 31 | \( 1 + (0.923 - 0.382i)T \) |
| 37 | \( 1 + (0.555 - 0.831i)T \) |
| 41 | \( 1 + (-0.555 - 0.831i)T \) |
| 43 | \( 1 + (-0.707 - 0.707i)T \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
| 53 | \( 1 + (-0.382 + 0.923i)T \) |
| 59 | \( 1 + (-0.831 - 0.555i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.195 + 0.980i)T \) |
| 71 | \( 1 + (0.555 - 0.831i)T \) |
| 73 | \( 1 + (0.707 - 0.707i)T \) |
| 79 | \( 1 + (-0.923 + 0.382i)T \) |
| 83 | \( 1 + (0.980 + 0.195i)T \) |
| 89 | \( 1 + (-0.382 - 0.923i)T \) |
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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
−24.65301524184210772617660830747, −23.62137926710351089493754612144, −22.6891838271818195115681355782, −21.76121512166719402066312374056, −21.47047931190147386632403546696, −20.415878702129176249820191069057, −19.41512751858380736448124645306, −18.464007923042282058581871730038, −17.15154977164880277529437639862, −16.771212224902641959139391697763, −15.99540700881627562303318209772, −14.912103694903279757573148154029, −13.84361579817898939320816986009, −13.193929652096686316565282718297, −11.87294585185958863739594983487, −11.006910059380322573515956011935, −9.900232351477456350981492589354, −9.442584228759126808351266227977, −8.5090396287786096323802827763, −6.65809641383736197654494044419, −6.06686441323958103229428237444, −5.09407113937548442360503691604, −3.83618652861589572712337376700, −2.9610307607025786508481230581, −1.20433177981546565095376708314,
0.95727021967038322577683384210, 2.29591583150108432897001977960, 3.17159976757825971299561915180, 5.09619335344253829489960872305, 5.86649718608098725542067433250, 6.808538637073175991389649638128, 7.5040050245913538916825386243, 9.009527531395440058672295956782, 9.81859031234725237663815875256, 10.783488654682534873840600734024, 12.04633795921607648557054051714, 12.76601080774018134919343246052, 13.4854711451044223849326370862, 14.35549011477986139144328073065, 15.548590767934353024307549355536, 16.68824492429243408746220362993, 17.415748603756462002960321396520, 18.2881217645213915470623218215, 18.8075560366488916948118774822, 20.04912554927534093933098884758, 20.666120210772091918137688963371, 22.3095287922049882786748009535, 22.45152473272045473334126890766, 23.29401716925386103734660016579, 24.693141042116448482906555733874