L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.959 + 0.281i)3-s + (0.415 + 0.909i)4-s + (0.654 − 0.755i)5-s + (−0.654 − 0.755i)6-s + (−0.841 − 0.540i)7-s + (0.142 − 0.989i)8-s + (0.841 + 0.540i)9-s + (−0.959 + 0.281i)10-s + (0.654 − 0.755i)11-s + (0.142 + 0.989i)12-s + (0.142 + 0.989i)13-s + (0.415 + 0.909i)14-s + (0.841 − 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.415 − 0.909i)17-s + ⋯ |
L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.959 + 0.281i)3-s + (0.415 + 0.909i)4-s + (0.654 − 0.755i)5-s + (−0.654 − 0.755i)6-s + (−0.841 − 0.540i)7-s + (0.142 − 0.989i)8-s + (0.841 + 0.540i)9-s + (−0.959 + 0.281i)10-s + (0.654 − 0.755i)11-s + (0.142 + 0.989i)12-s + (0.142 + 0.989i)13-s + (0.415 + 0.909i)14-s + (0.841 − 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.415 − 0.909i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.369629317 - 0.8376463815i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.369629317 - 0.8376463815i\) |
\(L(1)\) |
\(\approx\) |
\(1.063232802 - 0.3679573203i\) |
\(L(1)\) |
\(\approx\) |
\(1.063232802 - 0.3679573203i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 5 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.841 - 0.540i)T \) |
| 11 | \( 1 + (0.654 - 0.755i)T \) |
| 13 | \( 1 + (0.142 + 0.989i)T \) |
| 17 | \( 1 + (0.415 - 0.909i)T \) |
| 19 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (-0.959 - 0.281i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (0.142 - 0.989i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.415 + 0.909i)T \) |
| 43 | \( 1 + (-0.415 + 0.909i)T \) |
| 47 | \( 1 + (-0.959 - 0.281i)T \) |
| 53 | \( 1 + (-0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.142 + 0.989i)T \) |
| 61 | \( 1 + (0.654 + 0.755i)T \) |
| 71 | \( 1 + (0.415 + 0.909i)T \) |
| 73 | \( 1 + (-0.654 - 0.755i)T \) |
| 79 | \( 1 + (0.142 + 0.989i)T \) |
| 83 | \( 1 + (-0.654 + 0.755i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 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
−32.360855609139776713144466189678, −30.68655337467572371856212629536, −29.732862793376383446901037698879, −28.606045176923831917018275411672, −27.27499453791844828610910752162, −26.144012198886595400832644528211, −25.368722026209607383975156594930, −24.93869030513333261329575734465, −23.24803414945895139140024607750, −21.93173350239395427114623977815, −20.31729700077782076216896356666, −19.397797782536908294084838026016, −18.39871579573901798522031793921, −17.53521090274154772533362298393, −15.81902846070944133083293086876, −14.88495003957211474923834717920, −13.92105181215033523981360535707, −12.35885698415604089323787688383, −10.20548597142560955349836267785, −9.60092274802826390806516881558, −8.23302588550341353440006612856, −6.94827610406545022008786722861, −5.905277258695470374016831192888, −3.163944411479668328292701579668, −1.71271141402984651607222317689,
1.11213353390489316654576811079, 2.8001869815507215905817458218, 4.19389158159168737656219989232, 6.65238324211628244313151192729, 8.20632274005064085297210176401, 9.401632053809930023990079861934, 9.85403419803377776487027651945, 11.65959403220018419189921617405, 13.19433612770725507221100238741, 13.984260413990807351900781359122, 16.203208505768633499143494968014, 16.53723918024886531452560791078, 18.22569059147395980921757318087, 19.46634438073434232144064805640, 20.16509010923934783183278395325, 21.20452132978188128804676049241, 22.11570063505672744407846444110, 24.30940212456184537244247085736, 25.267401761481502331721946176462, 26.20072803986800816475524668981, 27.0135354339952858286590898906, 28.31005012253305362886951077238, 29.32118910740067237599383244506, 30.207362731825202059872977748731, 31.59738618834488549407400417931