L(s) = 1 | + (−0.327 + 0.945i)2-s + (−0.5 + 0.866i)3-s + (−0.786 − 0.618i)4-s + (0.0475 − 0.998i)5-s + (−0.654 − 0.755i)6-s + (0.841 − 0.540i)8-s + (−0.5 − 0.866i)9-s + (0.928 + 0.371i)10-s + (0.928 − 0.371i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.235 + 0.971i)16-s + (−0.995 − 0.0950i)17-s + (0.981 − 0.189i)18-s + (−0.995 + 0.0950i)19-s + (−0.654 + 0.755i)20-s + ⋯ |
L(s) = 1 | + (−0.327 + 0.945i)2-s + (−0.5 + 0.866i)3-s + (−0.786 − 0.618i)4-s + (0.0475 − 0.998i)5-s + (−0.654 − 0.755i)6-s + (0.841 − 0.540i)8-s + (−0.5 − 0.866i)9-s + (0.928 + 0.371i)10-s + (0.928 − 0.371i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.235 + 0.971i)16-s + (−0.995 − 0.0950i)17-s + (0.981 − 0.189i)18-s + (−0.995 + 0.0950i)19-s + (−0.654 + 0.755i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6307393104 + 0.04599695429i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6307393104 + 0.04599695429i\) |
\(L(1)\) |
\(\approx\) |
\(0.5898057579 + 0.2570590422i\) |
\(L(1)\) |
\(\approx\) |
\(0.5898057579 + 0.2570590422i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.327 + 0.945i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.0475 - 0.998i)T \) |
| 13 | \( 1 + (-0.142 + 0.989i)T \) |
| 17 | \( 1 + (-0.995 - 0.0950i)T \) |
| 19 | \( 1 + (-0.995 + 0.0950i)T \) |
| 23 | \( 1 + (0.235 + 0.971i)T \) |
| 29 | \( 1 + (0.415 - 0.909i)T \) |
| 31 | \( 1 + (0.928 + 0.371i)T \) |
| 37 | \( 1 + (-0.786 + 0.618i)T \) |
| 41 | \( 1 + (-0.654 - 0.755i)T \) |
| 43 | \( 1 + (0.841 - 0.540i)T \) |
| 47 | \( 1 + (0.981 + 0.189i)T \) |
| 53 | \( 1 + (0.235 - 0.971i)T \) |
| 59 | \( 1 + (-0.327 - 0.945i)T \) |
| 61 | \( 1 + (0.981 + 0.189i)T \) |
| 67 | \( 1 + (0.981 - 0.189i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.723 - 0.690i)T \) |
| 79 | \( 1 + (0.0475 - 0.998i)T \) |
| 83 | \( 1 + (-0.959 + 0.281i)T \) |
| 89 | \( 1 + (0.580 - 0.814i)T \) |
| 97 | \( 1 + (0.841 - 0.540i)T \) |
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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
−22.13266644919904721546801316666, −21.43661489672373816591661634098, −20.23152312121883474431157860889, −19.59740217961689601329081480081, −18.83042337846896754762724486201, −18.253897137440929055613473104762, −17.531395337242700896821954328693, −17.02511657666268175963895346163, −15.68751076096367520068749420390, −14.5848887555796492417683695713, −13.761844071012070559990129617637, −12.923518298117792681999765822, −12.34911586418430850730167543864, −11.32658580387190605501684406031, −10.71750461480842873848910332276, −10.203283841174660138635681320744, −8.79659839652853764791573846019, −8.055090665287738670962443709483, −7.08844202917296618970429159008, −6.32651837435882519859722372167, −5.161952603319494462426808698507, −4.03736796391905070092118364872, −2.700730159557852626762795375370, −2.32230559940125609958270873724, −0.93172174988561045233392344465,
0.43846212806859348492954404158, 1.88657648424133227690416156103, 3.86631878819038486886764699377, 4.54115330314468145048473252061, 5.20062638108713896521296309614, 6.16816955526348332437366676437, 6.91570906942336997405966403402, 8.28503794604953793477824452354, 8.91070459127439681515863584372, 9.56055161295669881943017814959, 10.404098553050819747938247439015, 11.44850708049121263332706426322, 12.3515966439005762780225593741, 13.46303034351165941364145778766, 14.15018136392200733091178032497, 15.41173833383815985368131184438, 15.64026535520596425618718553978, 16.55571241003320659578592610151, 17.33790705828990163864389868192, 17.48884499865422962978056200200, 18.92997099172428713577513241398, 19.6183564830663099631936843407, 20.68986492065249348302145906081, 21.40997381732650089353366412473, 22.1818450350357194836462550079