L(s) = 1 | + (−0.852 + 0.522i)3-s + (0.987 + 0.156i)7-s + (0.453 − 0.891i)9-s + (0.0784 − 0.996i)13-s + (0.309 − 0.951i)17-s + (−0.852 + 0.522i)19-s + (−0.923 + 0.382i)21-s + (−0.707 + 0.707i)23-s + (0.0784 + 0.996i)27-s + (0.972 − 0.233i)29-s + (−0.309 − 0.951i)31-s + (0.522 − 0.852i)37-s + (0.453 + 0.891i)39-s + (−0.156 − 0.987i)41-s + (0.382 + 0.923i)43-s + ⋯ |
L(s) = 1 | + (−0.852 + 0.522i)3-s + (0.987 + 0.156i)7-s + (0.453 − 0.891i)9-s + (0.0784 − 0.996i)13-s + (0.309 − 0.951i)17-s + (−0.852 + 0.522i)19-s + (−0.923 + 0.382i)21-s + (−0.707 + 0.707i)23-s + (0.0784 + 0.996i)27-s + (0.972 − 0.233i)29-s + (−0.309 − 0.951i)31-s + (0.522 − 0.852i)37-s + (0.453 + 0.891i)39-s + (−0.156 − 0.987i)41-s + (0.382 + 0.923i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.011725426 - 0.5711428501i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011725426 - 0.5711428501i\) |
\(L(1)\) |
\(\approx\) |
\(0.8857476185 + 0.01104049226i\) |
\(L(1)\) |
\(\approx\) |
\(0.8857476185 + 0.01104049226i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.852 + 0.522i)T \) |
| 7 | \( 1 + (0.987 + 0.156i)T \) |
| 13 | \( 1 + (0.0784 - 0.996i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.852 + 0.522i)T \) |
| 23 | \( 1 + (-0.707 + 0.707i)T \) |
| 29 | \( 1 + (0.972 - 0.233i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.522 - 0.852i)T \) |
| 41 | \( 1 + (-0.156 - 0.987i)T \) |
| 43 | \( 1 + (0.382 + 0.923i)T \) |
| 47 | \( 1 + (0.809 + 0.587i)T \) |
| 53 | \( 1 + (-0.760 + 0.649i)T \) |
| 59 | \( 1 + (-0.852 - 0.522i)T \) |
| 61 | \( 1 + (-0.649 + 0.760i)T \) |
| 67 | \( 1 + (-0.382 + 0.923i)T \) |
| 71 | \( 1 + (-0.453 - 0.891i)T \) |
| 73 | \( 1 + (0.156 - 0.987i)T \) |
| 79 | \( 1 + (0.951 - 0.309i)T \) |
| 83 | \( 1 + (0.649 - 0.760i)T \) |
| 89 | \( 1 + (0.707 - 0.707i)T \) |
| 97 | \( 1 + (0.951 - 0.309i)T \) |
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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
−18.66357081384280692142886644724, −18.1789916428056442236793544712, −17.3993416871456997320285404640, −16.948598215698612443333085605518, −16.32203461421425528743284106446, −15.44852859578357953174223008439, −14.61624722572773372063264917655, −13.984947853583717350455438397464, −13.32058748230293887935356328617, −12.32128706272752943829255175535, −12.06558209129802345588427529795, −11.0893110592964988979501096793, −10.72331310697606824013605982778, −9.96694744016981328964533938015, −8.761994978990099758575763396, −8.23648708267282077740119878277, −7.465163406623922985414753270341, −6.55473242773938264484844868086, −6.22139261521979184732192556385, −5.085479794180403593978128215637, −4.608922174727403828071142001406, −3.84673404933421095210816960658, −2.43025943655500547534001039334, −1.739590983205041220485852185300, −1.010313822017849931215137152952,
0.441492885663738150313160785654, 1.4134983675027844625638097421, 2.45279620045884008328563304844, 3.47875730475737280923654300544, 4.3729965715489745082615887858, 4.89364619474739399285533615245, 5.8471215001055517208358764985, 6.081396821673691205445366512686, 7.481748977870299129901957943166, 7.80460821403348711687923157721, 8.88927209027475525820826906680, 9.55961983977140710192846906164, 10.4782608442399698568486598434, 10.82051069283578455309877339688, 11.70972272389538806146186931434, 12.15119470994554424128478945771, 12.93249273425689703902856416454, 13.90824956091116263645922488608, 14.59816252755087389937366508029, 15.31190869357084965453297231284, 15.84080415906487935942542451518, 16.60221185836425911669446771316, 17.36197206358286148843692737906, 17.830983379930401788540947300765, 18.35837053899933838611275532456