L(s) = 1 | + (−0.540 − 0.841i)7-s + (−0.989 − 0.142i)11-s + (0.841 + 0.540i)13-s + (−0.281 + 0.959i)17-s + (0.281 + 0.959i)19-s + (0.281 − 0.959i)29-s + (0.415 + 0.909i)31-s + (−0.654 − 0.755i)37-s + (−0.654 + 0.755i)41-s + (−0.415 + 0.909i)43-s − i·47-s + (−0.415 + 0.909i)49-s + (0.841 − 0.540i)53-s + (0.540 − 0.841i)59-s + (−0.909 + 0.415i)61-s + ⋯ |
L(s) = 1 | + (−0.540 − 0.841i)7-s + (−0.989 − 0.142i)11-s + (0.841 + 0.540i)13-s + (−0.281 + 0.959i)17-s + (0.281 + 0.959i)19-s + (0.281 − 0.959i)29-s + (0.415 + 0.909i)31-s + (−0.654 − 0.755i)37-s + (−0.654 + 0.755i)41-s + (−0.415 + 0.909i)43-s − i·47-s + (−0.415 + 0.909i)49-s + (0.841 − 0.540i)53-s + (0.540 − 0.841i)59-s + (−0.909 + 0.415i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.001270001181 + 0.03900275792i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.001270001181 + 0.03900275792i\) |
\(L(1)\) |
\(\approx\) |
\(0.8261509487 + 0.01190171547i\) |
\(L(1)\) |
\(\approx\) |
\(0.8261509487 + 0.01190171547i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + (-0.540 - 0.841i)T \) |
| 11 | \( 1 + (-0.989 - 0.142i)T \) |
| 13 | \( 1 + (0.841 + 0.540i)T \) |
| 17 | \( 1 + (-0.281 + 0.959i)T \) |
| 19 | \( 1 + (0.281 + 0.959i)T \) |
| 29 | \( 1 + (0.281 - 0.959i)T \) |
| 31 | \( 1 + (0.415 + 0.909i)T \) |
| 37 | \( 1 + (-0.654 - 0.755i)T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + (-0.415 + 0.909i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.841 - 0.540i)T \) |
| 59 | \( 1 + (0.540 - 0.841i)T \) |
| 61 | \( 1 + (-0.909 + 0.415i)T \) |
| 67 | \( 1 + (0.142 + 0.989i)T \) |
| 71 | \( 1 + (-0.142 - 0.989i)T \) |
| 73 | \( 1 + (-0.281 - 0.959i)T \) |
| 79 | \( 1 + (-0.841 - 0.540i)T \) |
| 83 | \( 1 + (0.654 + 0.755i)T \) |
| 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
−17.689530638367963819403994917463, −16.86759342863633548610631060101, −15.95109859455780160241531961855, −15.6011794245028920104127883827, −15.24357002076559794152273754519, −14.147310978073427114231334842461, −13.38208948012879516213165453752, −13.090024210075831330335628834582, −12.19155903666850953908150076910, −11.64247694374933067407807108836, −10.815439176960541494592193746612, −10.22827278612551095615118422637, −9.42161192630411244256471047600, −8.78949992096985685422060849504, −8.22496686457825475273167117542, −7.27725710467309616592752772110, −6.71650727671967907899652440487, −5.78646278019163169366366576611, −5.28618617600721452471695443688, −4.58411401917896327899305874885, −3.45548708769566286101500955032, −2.83501228382849580520517816444, −2.28587622405096308739210892017, −1.09193273921313334637485107733, −0.01080846570291370657548159199,
1.189249816792236150232422123234, 1.93441108906551251467459764409, 3.01048720079070460292083359117, 3.67304931252562674516866020649, 4.27681010570204666083287528820, 5.18927099731530707089453393964, 6.04453858679823639404478282338, 6.56266861370040579172453368847, 7.35626170648312614121844187653, 8.166836944443262965712065728264, 8.5873993663804480742337493827, 9.64945812923568625974406885680, 10.26664201545103326339289600883, 10.68592630371096530831397504112, 11.49478234481712583617411829667, 12.285265509959351416941772902137, 13.04690440586273563592702488690, 13.52685112955194305603734523409, 14.06344660740020579081461557287, 14.95935930084593687778078870621, 15.63463266997375172928435608838, 16.38612438626226317625152657742, 16.611699760112265035653640551073, 17.63588025799211520502482059889, 18.104180081866597974520455829300