L(s) = 1 | + (−0.923 − 0.382i)2-s + (−0.382 − 0.923i)3-s + (0.707 + 0.707i)4-s + (−0.195 + 0.980i)5-s + i·6-s + (−0.195 − 0.980i)7-s + (−0.382 − 0.923i)8-s + (−0.707 + 0.707i)9-s + (0.555 − 0.831i)10-s + (0.382 + 0.923i)11-s + (0.382 − 0.923i)12-s + (0.980 + 0.195i)13-s + (−0.195 + 0.980i)14-s + (0.980 − 0.195i)15-s + i·16-s + (0.980 + 0.195i)17-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)2-s + (−0.382 − 0.923i)3-s + (0.707 + 0.707i)4-s + (−0.195 + 0.980i)5-s + i·6-s + (−0.195 − 0.980i)7-s + (−0.382 − 0.923i)8-s + (−0.707 + 0.707i)9-s + (0.555 − 0.831i)10-s + (0.382 + 0.923i)11-s + (0.382 − 0.923i)12-s + (0.980 + 0.195i)13-s + (−0.195 + 0.980i)14-s + (0.980 − 0.195i)15-s + i·16-s + (0.980 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0569 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0569 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6211692305 - 0.5867468437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6211692305 - 0.5867468437i\) |
\(L(1)\) |
\(\approx\) |
\(0.6094327547 - 0.2554863616i\) |
\(L(1)\) |
\(\approx\) |
\(0.6094327547 - 0.2554863616i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 3 | \( 1 + (-0.382 - 0.923i)T \) |
| 5 | \( 1 + (-0.195 + 0.980i)T \) |
| 7 | \( 1 + (-0.195 - 0.980i)T \) |
| 11 | \( 1 + (0.382 + 0.923i)T \) |
| 13 | \( 1 + (0.980 + 0.195i)T \) |
| 17 | \( 1 + (0.980 + 0.195i)T \) |
| 19 | \( 1 + (0.195 - 0.980i)T \) |
| 23 | \( 1 + (-0.555 - 0.831i)T \) |
| 29 | \( 1 + (-0.555 - 0.831i)T \) |
| 31 | \( 1 + (0.923 - 0.382i)T \) |
| 37 | \( 1 + (-0.831 - 0.555i)T \) |
| 41 | \( 1 + (0.831 - 0.555i)T \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 + (-0.707 - 0.707i)T \) |
| 53 | \( 1 + (0.382 - 0.923i)T \) |
| 59 | \( 1 + (0.555 - 0.831i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.980 + 0.195i)T \) |
| 71 | \( 1 + (0.831 + 0.555i)T \) |
| 73 | \( 1 + (0.707 - 0.707i)T \) |
| 79 | \( 1 + (-0.923 + 0.382i)T \) |
| 83 | \( 1 + (0.195 - 0.980i)T \) |
| 89 | \( 1 + (0.382 + 0.923i)T \) |
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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
−29.55562651331683428814250895936, −28.647661546938253832831156810, −27.7233878216499640769934447228, −27.426665508405233684334705522, −25.954927454219915875489900226819, −25.09549671983219237541670179806, −24.01958590409244735223717363247, −22.880517703983729817507344444398, −21.36304059571013437333142674350, −20.66413003383889936282597818179, −19.39729984934440245879525825453, −18.28048010619362155034425152023, −17.000109016060776869756137320, −16.141708059945099791291679004563, −15.64054747734687664329949863702, −14.22110449601010978668640096272, −12.18753616120558302106683471953, −11.31571628605456538717756499282, −9.91691787644001295100330399909, −8.94887585233905966376813565268, −8.191290066120684236635084913203, −6.00291998043824359820924863015, −5.40772432391866305426394741390, −3.457482682283633718322389274603, −1.10299785430130081647531325560,
0.68103282877061852673423668231, 2.20848681130587870882685553616, 3.77050842561568468052912132880, 6.38614994152791165881008951030, 7.13244894493714496094751696036, 8.08404440694484874369696192779, 9.868177906516226726156580268, 10.89472568093878590305073934875, 11.772595554975361653931988204314, 13.03399776169197732960304998876, 14.307489569081785897762117761611, 15.94180803145919064173004334701, 17.19182920966900010784141220637, 17.92069036697696304678515954146, 18.93781650484556894554563674275, 19.683571029051085915028326002318, 20.832236268721342366203195058669, 22.47847877235416184190924618194, 23.1927681787051092526021693985, 24.498475693013876318854458121021, 25.88902904524132178165730530325, 26.20403195262977942684213346305, 27.75032807921149559095742547604, 28.48153514068571705276653677686, 29.86449929226723979050917170059