L(s) = 1 | + i·3-s − i·7-s − 9-s − 11-s − i·13-s + i·17-s + 19-s + 21-s + i·23-s − i·27-s − 29-s − 31-s − i·33-s + i·37-s + 39-s + ⋯ |
L(s) = 1 | + i·3-s − i·7-s − 9-s − 11-s − i·13-s + i·17-s + 19-s + 21-s + i·23-s − i·27-s − 29-s − 31-s − i·33-s + i·37-s + 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6153549914 + 0.1748094575i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6153549914 + 0.1748094575i\) |
\(L(1)\) |
\(\approx\) |
\(0.8595478356 + 0.1767421806i\) |
\(L(1)\) |
\(\approx\) |
\(0.8595478356 + 0.1767421806i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 \) |
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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
−40.38883430358204640546448355975, −38.735795968667743215521103873541, −37.247801206047243723799357576921, −36.09441314324750012996248414038, −34.856507365903759726131828355361, −33.78567981564399359465343199462, −31.6515966356349751302737680839, −30.91844193420268949843068014374, −29.237289040625714209614816587000, −28.36597353127179346163501353088, −26.33239523022640549254450971302, −24.963380929599044063195818592980, −23.8957002429788890790730956947, −22.38526032224391234553804965253, −20.64653162258021723690996015926, −18.88172009903748921865918078907, −18.10099096196119455815502387985, −16.15266427433562288425121751476, −14.28943082233952163374769888088, −12.7635194118700229837484632365, −11.459596005614086402668986184279, −9.03603034832864075846403276009, −7.35262170860794421983846811052, −5.554135368747861766165490725370, −2.43970471075016573834496104089,
3.53109568390976713881226562799, 5.37092217269348812234929796648, 7.86202871799724463684935052995, 9.885690281516151356044990603413, 11.001014977110963206286226455439, 13.27477102305485924531921767602, 14.96782345013431223635605146029, 16.30388879911594151964725727174, 17.69750908991989790359894140769, 19.882153214576594928835531512624, 20.93797208528347140576261539997, 22.434540191315472933521500272742, 23.7243438948056873765175022385, 25.77843933969612115054892329945, 26.78068812058721427987306146909, 28.047119757788737551702032536619, 29.504789230302340497825221713429, 31.1632075039371300308856175534, 32.570556274132070632133243141970, 33.45096533960072786896162409934, 34.849288868370123408737536992514, 36.63925162106026851035280910428, 37.64121840189302633480402058127, 39.336264703355623494329443153673, 39.68269158589923208615540280166