L(s) = 1 | + (−0.923 − 0.382i)3-s + (−0.555 + 0.831i)5-s + (0.555 + 0.831i)7-s + (0.707 + 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.831 − 0.555i)13-s + (0.831 − 0.555i)15-s + (−0.831 − 0.555i)17-s + (−0.555 + 0.831i)19-s + (−0.195 − 0.980i)21-s + (0.980 − 0.195i)23-s + (−0.382 − 0.923i)25-s + (−0.382 − 0.923i)27-s + (−0.980 + 0.195i)29-s + (−0.382 + 0.923i)31-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)3-s + (−0.555 + 0.831i)5-s + (0.555 + 0.831i)7-s + (0.707 + 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.831 − 0.555i)13-s + (0.831 − 0.555i)15-s + (−0.831 − 0.555i)17-s + (−0.555 + 0.831i)19-s + (−0.195 − 0.980i)21-s + (0.980 − 0.195i)23-s + (−0.382 − 0.923i)25-s + (−0.382 − 0.923i)27-s + (−0.980 + 0.195i)29-s + (−0.382 + 0.923i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2688565888 + 0.5143908059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2688565888 + 0.5143908059i\) |
\(L(1)\) |
\(\approx\) |
\(0.6552173097 + 0.1919048292i\) |
\(L(1)\) |
\(\approx\) |
\(0.6552173097 + 0.1919048292i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 97 | \( 1 \) |
good | 3 | \( 1 + (-0.923 - 0.382i)T \) |
| 5 | \( 1 + (-0.555 + 0.831i)T \) |
| 7 | \( 1 + (0.555 + 0.831i)T \) |
| 11 | \( 1 + (0.923 + 0.382i)T \) |
| 13 | \( 1 + (-0.831 - 0.555i)T \) |
| 17 | \( 1 + (-0.831 - 0.555i)T \) |
| 19 | \( 1 + (-0.555 + 0.831i)T \) |
| 23 | \( 1 + (0.980 - 0.195i)T \) |
| 29 | \( 1 + (-0.980 + 0.195i)T \) |
| 31 | \( 1 + (-0.382 + 0.923i)T \) |
| 37 | \( 1 + (-0.195 + 0.980i)T \) |
| 41 | \( 1 + (0.195 + 0.980i)T \) |
| 43 | \( 1 + (0.707 - 0.707i)T \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (-0.923 + 0.382i)T \) |
| 59 | \( 1 + (-0.980 - 0.195i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.831 + 0.555i)T \) |
| 71 | \( 1 + (-0.195 + 0.980i)T \) |
| 73 | \( 1 + (-0.707 - 0.707i)T \) |
| 79 | \( 1 + (0.382 - 0.923i)T \) |
| 83 | \( 1 + (-0.555 + 0.831i)T \) |
| 89 | \( 1 + (-0.923 - 0.382i)T \) |
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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
−24.260265134502222745951114888283, −23.39241145163791534729367988309, −22.47836347536735234178024939313, −21.59161106290909379516428978927, −20.80528328790693178091641119470, −19.8113567497754450883855009306, −19.114327008919513401359754857158, −17.568726480055054023646558389400, −17.073270524583163744170639827422, −16.534063292768072632961902951966, −15.41028160137781331049286271731, −14.59077651632923693271808613111, −13.25418678662537818103782098972, −12.4430202237785403458719689058, −11.23094597032233743736950229262, −11.12599629180786250300493873386, −9.5666748391543063393390300550, −8.8388025667153338997241643561, −7.47186600003704884132494336112, −6.64704173380419575957932231836, −5.331395492810487727347995871002, −4.39379454807929940686436936796, −3.89460702501632005971470896750, −1.69023872327508107220756256783, −0.39982082936253546724221713370,
1.61166250058357259038952974089, 2.7967094303446408998536642550, 4.32209793368566177725640645571, 5.25680823444241898736045365799, 6.41738790283261829382661694657, 7.145614425122167177059326757953, 8.1199394279185848618807643142, 9.42684478039039213126176761376, 10.642475859662736053089087519692, 11.37118434740584524693290742755, 12.10073867157203128486717902929, 12.85388901829744053682279477200, 14.37930999785587026568787077469, 14.99313961184253026691155144625, 15.91609755767243958826672263322, 17.1018036391614750749473356466, 17.76134759709784476953416167869, 18.63566931272396492599808120349, 19.24673177473486431367458366808, 20.35923944075234055710721331810, 21.69773365732059608740184906992, 22.33119332274791029410806727530, 22.86019864191770726603055251289, 23.87132037293663530469455278393, 24.77450223403287962262437115507