L(s) = 1 | + (−0.760 + 0.649i)3-s + (0.707 + 0.707i)7-s + (0.156 − 0.987i)9-s + (−0.852 + 0.522i)11-s + (−0.522 + 0.852i)13-s + (0.951 + 0.309i)17-s + (−0.0784 − 0.996i)19-s + (−0.996 − 0.0784i)21-s + (0.987 − 0.156i)23-s + (0.522 + 0.852i)27-s + (−0.760 + 0.649i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.972 − 0.233i)37-s + (−0.156 − 0.987i)39-s + ⋯ |
L(s) = 1 | + (−0.760 + 0.649i)3-s + (0.707 + 0.707i)7-s + (0.156 − 0.987i)9-s + (−0.852 + 0.522i)11-s + (−0.522 + 0.852i)13-s + (0.951 + 0.309i)17-s + (−0.0784 − 0.996i)19-s + (−0.996 − 0.0784i)21-s + (0.987 − 0.156i)23-s + (0.522 + 0.852i)27-s + (−0.760 + 0.649i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.972 − 0.233i)37-s + (−0.156 − 0.987i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2666073127 + 0.8727141511i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2666073127 + 0.8727141511i\) |
\(L(1)\) |
\(\approx\) |
\(0.7419067313 + 0.3532012337i\) |
\(L(1)\) |
\(\approx\) |
\(0.7419067313 + 0.3532012337i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.760 + 0.649i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (-0.852 + 0.522i)T \) |
| 13 | \( 1 + (-0.522 + 0.852i)T \) |
| 17 | \( 1 + (0.951 + 0.309i)T \) |
| 19 | \( 1 + (-0.0784 - 0.996i)T \) |
| 23 | \( 1 + (0.987 - 0.156i)T \) |
| 29 | \( 1 + (-0.760 + 0.649i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.972 - 0.233i)T \) |
| 41 | \( 1 + (-0.156 + 0.987i)T \) |
| 43 | \( 1 + (0.382 - 0.923i)T \) |
| 47 | \( 1 + (-0.951 + 0.309i)T \) |
| 53 | \( 1 + (-0.996 - 0.0784i)T \) |
| 59 | \( 1 + (0.972 - 0.233i)T \) |
| 61 | \( 1 + (-0.233 + 0.972i)T \) |
| 67 | \( 1 + (0.996 - 0.0784i)T \) |
| 71 | \( 1 + (0.453 - 0.891i)T \) |
| 73 | \( 1 + (-0.987 + 0.156i)T \) |
| 79 | \( 1 + (-0.951 + 0.309i)T \) |
| 83 | \( 1 + (0.0784 + 0.996i)T \) |
| 89 | \( 1 + (-0.156 - 0.987i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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
−20.29855178202891753046077124386, −19.144260160132011279124117639923, −18.67676529873520528060505518413, −17.93692516753898094863977629665, −17.123395838572425827798297839189, −16.732985845722966102562040334251, −15.85212983196459276973640261573, −14.79749142546979619680519303555, −14.13991830259500496389470261543, −13.1069146517470802206269893789, −12.83814380966699967652260582954, −11.69658232373881602354580734505, −11.157262722128790314759072640825, −10.394477872182741369341866300597, −9.730179063321785665407100451808, −8.17278592451460776638761667946, −7.79482562476043222906824362081, −7.178409193694421699111358408413, −5.95044798192019926639794535743, −5.41111770927538228877265808105, −4.65120462693476366201930990911, −3.46083489331559417790938447600, −2.39364269653174029572754022206, −1.32205136928474871777955678251, −0.41201745792413161208143466524,
1.250304354033102478225439336176, 2.363879387737659837705750916325, 3.35573521428985480328593335126, 4.62881506838750321393069087555, 4.98316306022434025521229588062, 5.74424620698571981393469006526, 6.80135588346784086821002581999, 7.55959861048720919031594107896, 8.68093200516699261316133036285, 9.369640140856710083583813758627, 10.140442412739766257670769408942, 11.05688603550384682818653064397, 11.50175218088351258485352707655, 12.44409799150821464656590791804, 12.94518785765199680647714067118, 14.37433308583505590607816298369, 14.84380781703109688452035433065, 15.54148218178202892585923844976, 16.34274458325899784683562874776, 17.02895420032935114864335911241, 17.79139755497115696586408442761, 18.37369100548763800400418120204, 19.150318273298440942321274226057, 20.22711711096053855176575889492, 21.05937466553930875540688046063