L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − i·4-s + (−0.382 − 0.923i)5-s − 6-s + (−0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + i·9-s + (0.923 + 0.382i)10-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s + (0.382 + 0.923i)13-s + (−0.382 − 0.923i)14-s + (0.382 − 0.923i)15-s − 16-s + (0.382 + 0.923i)17-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − i·4-s + (−0.382 − 0.923i)5-s − 6-s + (−0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + i·9-s + (0.923 + 0.382i)10-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s + (0.382 + 0.923i)13-s + (−0.382 − 0.923i)14-s + (0.382 − 0.923i)15-s − 16-s + (0.382 + 0.923i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5468672172 + 0.6124708509i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5468672172 + 0.6124708509i\) |
\(L(1)\) |
\(\approx\) |
\(0.7389995239 + 0.4514630738i\) |
\(L(1)\) |
\(\approx\) |
\(0.7389995239 + 0.4514630738i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.382 - 0.923i)T \) |
| 7 | \( 1 + (-0.382 + 0.923i)T \) |
| 11 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + (0.382 + 0.923i)T \) |
| 17 | \( 1 + (0.382 + 0.923i)T \) |
| 19 | \( 1 + (-0.382 - 0.923i)T \) |
| 23 | \( 1 + (0.923 - 0.382i)T \) |
| 29 | \( 1 + (0.923 - 0.382i)T \) |
| 31 | \( 1 + (-0.707 - 0.707i)T \) |
| 37 | \( 1 + (-0.923 - 0.382i)T \) |
| 41 | \( 1 + (-0.923 + 0.382i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.707 - 0.707i)T \) |
| 59 | \( 1 + (0.923 + 0.382i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.382 + 0.923i)T \) |
| 71 | \( 1 + (-0.923 - 0.382i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.707 - 0.707i)T \) |
| 83 | \( 1 + (-0.382 - 0.923i)T \) |
| 89 | \( 1 + (0.707 + 0.707i)T \) |
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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
−29.624695197084017894178411622956, −29.39667179914159584429856774604, −27.28809108008035273326987124818, −26.97652353281210637404118206344, −25.7319473986132097211395255230, −25.09169178433631472680192126065, −23.37519224836013014326003599473, −22.511790680159391254709447734538, −21.01074261707408316441092709462, −19.981829346212419897247106693786, −19.24535208131778315362174473848, −18.44931110759888941210350902346, −17.36720305281047714266309272878, −16.00704805742049875398491183936, −14.3955044817768849774751214174, −13.464308576349751185568601801834, −12.237496679174009640151254333738, −11.02600169889289495889321079860, −9.976494260653600296376293492399, −8.58020245632630691121972181618, −7.50900325100954554015083243049, −6.63603505904807436097936057252, −3.61994440142703590956172682903, −3.04170251869705487163445011786, −1.11957195672536038071649626917,
1.94227475583114302831038365624, 4.116106440910398916183212906288, 5.30768987175230816170642716820, 6.90973259454634705975935195360, 8.596878160854882804773013870906, 8.90085960503842699538344251132, 10.02482792055360428397519922452, 11.63317863703960888316269426564, 13.18883479574082047026845296335, 14.69739823142842460285531624069, 15.41914490857494887064855717952, 16.36371602574265864105586690497, 17.24839379453181227644584013783, 18.97897659434291207121241743715, 19.565054959622627805127952884, 20.689224093550347184479657109540, 21.87904286730920386072160154030, 23.31294725252268036129649043173, 24.50000794639883848646981412749, 25.34390875323556899276849999789, 26.076506882598696663545100675103, 27.309704432852860317972191187106, 28.13363311360398328431372232358, 28.63504213677375978399141972526, 30.734408999232075980790504921667