L(s) = 1 | + (0.281 + 0.959i)3-s + (−0.989 + 0.142i)7-s + (−0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (−0.989 − 0.142i)13-s + (0.755 − 0.654i)17-s + (0.654 − 0.755i)19-s + (−0.415 − 0.909i)21-s + (−0.755 − 0.654i)27-s + (0.654 + 0.755i)29-s + (0.959 + 0.281i)31-s + (0.989 + 0.142i)33-s + (0.540 + 0.841i)37-s + (−0.142 − 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯ |
L(s) = 1 | + (0.281 + 0.959i)3-s + (−0.989 + 0.142i)7-s + (−0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (−0.989 − 0.142i)13-s + (0.755 − 0.654i)17-s + (0.654 − 0.755i)19-s + (−0.415 − 0.909i)21-s + (−0.755 − 0.654i)27-s + (0.654 + 0.755i)29-s + (0.959 + 0.281i)31-s + (0.989 + 0.142i)33-s + (0.540 + 0.841i)37-s + (−0.142 − 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.672698149 + 0.5786273413i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.672698149 + 0.5786273413i\) |
\(L(1)\) |
\(\approx\) |
\(1.038762799 + 0.2657667830i\) |
\(L(1)\) |
\(\approx\) |
\(1.038762799 + 0.2657667830i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.281 + 0.959i)T \) |
| 7 | \( 1 + (-0.989 + 0.142i)T \) |
| 11 | \( 1 + (0.415 - 0.909i)T \) |
| 13 | \( 1 + (-0.989 - 0.142i)T \) |
| 17 | \( 1 + (0.755 - 0.654i)T \) |
| 19 | \( 1 + (0.654 - 0.755i)T \) |
| 29 | \( 1 + (0.654 + 0.755i)T \) |
| 31 | \( 1 + (0.959 + 0.281i)T \) |
| 37 | \( 1 + (0.540 + 0.841i)T \) |
| 41 | \( 1 + (0.841 + 0.540i)T \) |
| 43 | \( 1 + (-0.281 - 0.959i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.989 + 0.142i)T \) |
| 59 | \( 1 + (-0.142 + 0.989i)T \) |
| 61 | \( 1 + (0.959 + 0.281i)T \) |
| 67 | \( 1 + (0.909 - 0.415i)T \) |
| 71 | \( 1 + (-0.415 - 0.909i)T \) |
| 73 | \( 1 + (-0.755 - 0.654i)T \) |
| 79 | \( 1 + (0.142 - 0.989i)T \) |
| 83 | \( 1 + (0.540 + 0.841i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 97 | \( 1 + (-0.540 + 0.841i)T \) |
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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
−23.46411712941752621310790808411, −22.96821639060971946511774792811, −22.12649437465677435899200209803, −20.89780611855202168398487895580, −19.925649864642823913939060926174, −19.382648278217513008509181466032, −18.66855656523392607558243255150, −17.53562237281380836379869305287, −16.97032147250958702521698086353, −15.83601995515481199216415158063, −14.67928553472693120916904674658, −14.10074044394614573389753798919, −12.90899352676841058523457910266, −12.41194705646683584830323221330, −11.627111683543770347855198464546, −9.99867929256463747393317896333, −9.55951549417317529611318001358, −8.20379297643469083832875677805, −7.3675758122037651280886640353, −6.57367562164009469602592040438, −5.65408643573791941225015918433, −4.15163983125165123809726310805, −3.0236992513035333882314772704, −2.00031590296190652168789221495, −0.72589399543208133427144507195,
0.69487781192600185122870524554, 2.811795356755947071932879576010, 3.2018692238886873185347132647, 4.54686938640729618181983731558, 5.46529859651639825161442818557, 6.53313636003432536768097273218, 7.74515389626579238504079846669, 8.90602257255531666455227265714, 9.58654722503649677121362911485, 10.314128249766203347640064691507, 11.43782932520750429857251580889, 12.293477475042335706513344203404, 13.56387258224668589030842824306, 14.2396041851469522181366430578, 15.23600171358142600322475208842, 16.1161455080712655626058325511, 16.59644872478978023152987465737, 17.62376131801798364076138618764, 18.96953635551042499812845605282, 19.57447305699948429660081304952, 20.343629375970644646889804927409, 21.40168641696689131006708784188, 22.12349080242185791228244053200, 22.59343836988278652178100889018, 23.75340217775719468626411824753