L(s) = 1 | + (−0.540 + 0.841i)7-s + (0.142 + 0.989i)11-s + (−0.540 − 0.841i)13-s + (−0.281 − 0.959i)17-s + (0.959 + 0.281i)19-s + (−0.959 + 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.755 + 0.654i)37-s + (0.654 + 0.755i)41-s + (0.909 − 0.415i)43-s − i·47-s + (−0.415 − 0.909i)49-s + (0.540 − 0.841i)53-s + (−0.841 + 0.540i)59-s + (−0.415 + 0.909i)61-s + ⋯ |
L(s) = 1 | + (−0.540 + 0.841i)7-s + (0.142 + 0.989i)11-s + (−0.540 − 0.841i)13-s + (−0.281 − 0.959i)17-s + (0.959 + 0.281i)19-s + (−0.959 + 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.755 + 0.654i)37-s + (0.654 + 0.755i)41-s + (0.909 − 0.415i)43-s − i·47-s + (−0.415 − 0.909i)49-s + (0.540 − 0.841i)53-s + (−0.841 + 0.540i)59-s + (−0.415 + 0.909i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4130541376 + 0.7631294392i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4130541376 + 0.7631294392i\) |
\(L(1)\) |
\(\approx\) |
\(0.8609699359 + 0.2066662543i\) |
\(L(1)\) |
\(\approx\) |
\(0.8609699359 + 0.2066662543i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + (-0.540 + 0.841i)T \) |
| 11 | \( 1 + (0.142 + 0.989i)T \) |
| 13 | \( 1 + (-0.540 - 0.841i)T \) |
| 17 | \( 1 + (-0.281 - 0.959i)T \) |
| 19 | \( 1 + (0.959 + 0.281i)T \) |
| 29 | \( 1 + (-0.959 + 0.281i)T \) |
| 31 | \( 1 + (-0.415 + 0.909i)T \) |
| 37 | \( 1 + (0.755 + 0.654i)T \) |
| 41 | \( 1 + (0.654 + 0.755i)T \) |
| 43 | \( 1 + (0.909 - 0.415i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.540 - 0.841i)T \) |
| 59 | \( 1 + (-0.841 + 0.540i)T \) |
| 61 | \( 1 + (-0.415 + 0.909i)T \) |
| 67 | \( 1 + (-0.989 - 0.142i)T \) |
| 71 | \( 1 + (-0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.281 + 0.959i)T \) |
| 79 | \( 1 + (-0.841 + 0.540i)T \) |
| 83 | \( 1 + (-0.755 - 0.654i)T \) |
| 89 | \( 1 + (-0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.755 + 0.654i)T \) |
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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
−20.49051286700083198519862718165, −19.72159881100552552156316521170, −19.19560416023041822934846830176, −18.42537372785454574178856271, −17.376435379622927199577811575832, −16.71984351548957832286933210173, −16.24395495132303903060075463013, −15.25560105741428796498849915958, −14.35075580955104795362698595675, −13.68049648840913128027699034129, −13.04820031127909793595634416600, −12.10922791553629336624877333705, −11.17560372109552953732988708375, −10.65138294895393422863402586863, −9.522351239118220428993501777089, −9.1032171810490438810452532818, −7.860322794731426341369772739459, −7.250368528973715990646472120630, −6.29002161705044814497980753939, −5.60723233861545459702461988814, −4.30881237800900182406374754583, −3.75929356958828907779968956802, −2.73117577741654009629566327319, −1.56582739122309688421164185246, −0.34133916128705142036881980336,
1.30032895118670656685935338193, 2.53754199257012063540416912589, 3.08734081082025611610339064985, 4.35139253580872450451191607949, 5.257072817968205562416875168598, 5.90893677990440392301628463851, 7.08015921094345742192927519811, 7.57528904197355406878724163858, 8.74310773338582615535747682078, 9.54163239682982290265739597442, 9.99299221641903575047270752405, 11.15340749083319281185352381647, 12.00243554933808395540091541823, 12.59304654160925732551948059999, 13.28935772759046577683037048406, 14.39760122394989457406050602251, 15.01513275581758661429606879124, 15.79566598096909647469841968187, 16.389184082530437465000035112949, 17.47487330415969494984334149016, 18.10932415740720489514184118549, 18.68375540129966226104103019151, 19.79104642251461153387803139276, 20.16794697086473170971154149398, 21.04429106392629462141305229473