L(s) = 1 | + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (−0.980 − 0.195i)5-s + (0.980 − 0.195i)7-s + (−0.382 − 0.923i)8-s + (0.831 + 0.555i)10-s + (0.382 + 0.923i)11-s + (0.195 − 0.980i)13-s + (−0.980 − 0.195i)14-s + i·16-s + (−0.195 + 0.980i)17-s + (−0.980 − 0.195i)19-s + (−0.555 − 0.831i)20-s − i·22-s + (0.831 − 0.555i)23-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (−0.980 − 0.195i)5-s + (0.980 − 0.195i)7-s + (−0.382 − 0.923i)8-s + (0.831 + 0.555i)10-s + (0.382 + 0.923i)11-s + (0.195 − 0.980i)13-s + (−0.980 − 0.195i)14-s + i·16-s + (−0.195 + 0.980i)17-s + (−0.980 − 0.195i)19-s + (−0.555 − 0.831i)20-s − i·22-s + (0.831 − 0.555i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7435682556 - 0.1921142398i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7435682556 - 0.1921142398i\) |
\(L(1)\) |
\(\approx\) |
\(0.7010713024 - 0.1223905231i\) |
\(L(1)\) |
\(\approx\) |
\(0.7010713024 - 0.1223905231i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 97 | \( 1 \) |
good | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 5 | \( 1 + (-0.980 - 0.195i)T \) |
| 7 | \( 1 + (0.980 - 0.195i)T \) |
| 11 | \( 1 + (0.382 + 0.923i)T \) |
| 13 | \( 1 + (0.195 - 0.980i)T \) |
| 17 | \( 1 + (-0.195 + 0.980i)T \) |
| 19 | \( 1 + (-0.980 - 0.195i)T \) |
| 23 | \( 1 + (0.831 - 0.555i)T \) |
| 29 | \( 1 + (0.831 - 0.555i)T \) |
| 31 | \( 1 + (-0.923 + 0.382i)T \) |
| 37 | \( 1 + (0.555 - 0.831i)T \) |
| 41 | \( 1 + (0.555 + 0.831i)T \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.382 - 0.923i)T \) |
| 59 | \( 1 + (-0.831 - 0.555i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.195 - 0.980i)T \) |
| 71 | \( 1 + (0.555 - 0.831i)T \) |
| 73 | \( 1 + (0.707 - 0.707i)T \) |
| 79 | \( 1 + (0.923 - 0.382i)T \) |
| 83 | \( 1 + (0.980 + 0.195i)T \) |
| 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
−25.614485012825825536885269542448, −24.64614342255594013521361358585, −23.86499060247906492472222340618, −23.31455433381170901469923877316, −21.82322960001768230465682843297, −20.84827005705342724798030481085, −19.9011919397899842817365752749, −18.91164105998659297543676794822, −18.523530603332091686474861100689, −17.26473251148270057829420383405, −16.46130416486050330800081194986, −15.58874524580456116132609659561, −14.697090431331018086220165567860, −13.90999094380953812311743153889, −12.04585357201720648230880087032, −11.305512545637677997146331709075, −10.75007915614373297716596556578, −9.08496304639324316177999089093, −8.559908223907132024366367000114, −7.50020535821720879996798561328, −6.66952685286276308939808492761, −5.33982749662041181108071590409, −4.07259205438334380982966438306, −2.498862292994496091181421001095, −1.03007614415778791379515071488,
0.959691766366157419292642662570, 2.28074580374328552193900148042, 3.76357921609158900399254833742, 4.696119306485185312902283472904, 6.50336170376812968234098480370, 7.67025022098334197340646565182, 8.21648200855691264844160073374, 9.204638139603361823713125255015, 10.67287677744394029828602810377, 11.03988515249755534395795334144, 12.30239200402562849870057253247, 12.841020430675962509794076155092, 14.78198200342698920170778417, 15.26398514085112321132194679985, 16.45819133689806037990708140601, 17.41896844809832753562216433159, 17.98566983332233302989127408771, 19.23666602848079609377273157867, 19.87225347471552283064496528460, 20.64388857149684293717213345142, 21.47428175551075294938228132497, 22.78306435977781250234855401748, 23.67540512661131815643315953299, 24.69094966146078992039449331078, 25.46357762442980895201175883299