L(s) = 1 | + (0.669 − 0.743i)2-s + (0.968 − 0.248i)3-s + (−0.104 − 0.994i)4-s + (−0.855 − 0.518i)5-s + (0.463 − 0.886i)6-s + (−0.756 + 0.653i)7-s + (−0.809 − 0.587i)8-s + (0.876 − 0.481i)9-s + (−0.957 + 0.289i)10-s + (−0.268 − 0.963i)11-s + (−0.348 − 0.937i)12-s + (0.783 − 0.621i)13-s + (−0.0209 + 0.999i)14-s + (−0.957 − 0.289i)15-s + (−0.978 + 0.207i)16-s + (0.944 − 0.328i)17-s + ⋯ |
L(s) = 1 | + (0.669 − 0.743i)2-s + (0.968 − 0.248i)3-s + (−0.104 − 0.994i)4-s + (−0.855 − 0.518i)5-s + (0.463 − 0.886i)6-s + (−0.756 + 0.653i)7-s + (−0.809 − 0.587i)8-s + (0.876 − 0.481i)9-s + (−0.957 + 0.289i)10-s + (−0.268 − 0.963i)11-s + (−0.348 − 0.937i)12-s + (0.783 − 0.621i)13-s + (−0.0209 + 0.999i)14-s + (−0.957 − 0.289i)15-s + (−0.978 + 0.207i)16-s + (0.944 − 0.328i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9068857803 - 1.411573620i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9068857803 - 1.411573620i\) |
\(L(1)\) |
\(\approx\) |
\(1.229183774 - 0.9620937477i\) |
\(L(1)\) |
\(\approx\) |
\(1.229183774 - 0.9620937477i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 151 | \( 1 \) |
good | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 3 | \( 1 + (0.968 - 0.248i)T \) |
| 5 | \( 1 + (-0.855 - 0.518i)T \) |
| 7 | \( 1 + (-0.756 + 0.653i)T \) |
| 11 | \( 1 + (-0.268 - 0.963i)T \) |
| 13 | \( 1 + (0.783 - 0.621i)T \) |
| 17 | \( 1 + (0.944 - 0.328i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.913 + 0.406i)T \) |
| 29 | \( 1 + (0.0627 + 0.998i)T \) |
| 31 | \( 1 + (0.604 + 0.796i)T \) |
| 37 | \( 1 + (0.146 + 0.989i)T \) |
| 41 | \( 1 + (0.535 + 0.844i)T \) |
| 43 | \( 1 + (-0.756 - 0.653i)T \) |
| 47 | \( 1 + (-0.999 + 0.0418i)T \) |
| 53 | \( 1 + (-0.637 - 0.770i)T \) |
| 59 | \( 1 + (0.309 - 0.951i)T \) |
| 61 | \( 1 + (-0.699 + 0.714i)T \) |
| 67 | \( 1 + (0.876 + 0.481i)T \) |
| 71 | \( 1 + (0.944 + 0.328i)T \) |
| 73 | \( 1 + (-0.187 - 0.982i)T \) |
| 79 | \( 1 + (-0.992 - 0.125i)T \) |
| 83 | \( 1 + (-0.425 - 0.904i)T \) |
| 89 | \( 1 + (0.996 + 0.0836i)T \) |
| 97 | \( 1 + (-0.0209 - 0.999i)T \) |
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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
−28.120732609008625312651388996257, −26.945091638326249988478614808057, −25.97739587783901564835743259794, −25.85313605553523605616684717757, −24.468740507976134513970200389305, −23.20663774339076207509214843819, −22.94480969304744796305369427508, −21.46645930558415065319335005171, −20.62382608952211514004363864453, −19.50916381833465026847501750912, −18.61148200742925879838278410667, −17.00656031009008108401297088568, −15.95605914783265464881870210406, −15.23019424918544597347887854646, −14.404386270935941253300649359146, −13.336059384661801555691981571738, −12.47546464621644823903343874540, −10.92290036303000691079999292043, −9.56759449543155841870997760016, −8.25156982129878285718800096561, −7.35484255263644748373341784777, −6.50039068998503803564635479459, −4.46741747347173207516622312548, −3.78937698897102550395530467657, −2.66642879735267794832654523014,
1.20277434799558850958564721180, 3.08934910846040420103923480590, 3.47997232092356230723039992875, 5.11506281754459391009469114635, 6.47195730882199940014944102005, 8.17657507846310875471955882278, 8.97633057862039544038602656934, 10.24495498304996481331496735462, 11.60286303141577591760064085925, 12.69801113090871638091037142663, 13.20561601230763778337893656406, 14.4962683869559282286359640204, 15.473920926209527680479334364898, 16.23476028026283953285648968278, 18.52328394618785112661071953629, 19.03247245099324180840215878149, 19.837899182035809397340955777801, 20.83551962498309151102483349509, 21.53553660188532690477483737452, 22.98369004247506696810093770227, 23.65380484513160028864503292476, 24.78390777933238973298831828332, 25.5356325142389460023457533586, 27.06695373845937261947289731667, 27.747064825672072170370023848142