L(s) = 1 | + (−0.974 − 0.222i)3-s + (−0.623 + 0.781i)5-s + (0.222 − 0.974i)7-s + (0.900 + 0.433i)9-s + (−0.433 − 0.900i)11-s + (0.900 − 0.433i)13-s + (0.781 − 0.623i)15-s − i·17-s + (0.974 − 0.222i)19-s + (−0.433 + 0.900i)21-s + (−0.623 − 0.781i)23-s + (−0.222 − 0.974i)25-s + (−0.781 − 0.623i)27-s + (0.781 + 0.623i)31-s + (0.222 + 0.974i)33-s + ⋯ |
L(s) = 1 | + (−0.974 − 0.222i)3-s + (−0.623 + 0.781i)5-s + (0.222 − 0.974i)7-s + (0.900 + 0.433i)9-s + (−0.433 − 0.900i)11-s + (0.900 − 0.433i)13-s + (0.781 − 0.623i)15-s − i·17-s + (0.974 − 0.222i)19-s + (−0.433 + 0.900i)21-s + (−0.623 − 0.781i)23-s + (−0.222 − 0.974i)25-s + (−0.781 − 0.623i)27-s + (0.781 + 0.623i)31-s + (0.222 + 0.974i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5765378685 - 0.3417084244i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5765378685 - 0.3417084244i\) |
\(L(1)\) |
\(\approx\) |
\(0.7206481340 - 0.1587888827i\) |
\(L(1)\) |
\(\approx\) |
\(0.7206481340 - 0.1587888827i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (-0.974 - 0.222i)T \) |
| 5 | \( 1 + (-0.623 + 0.781i)T \) |
| 7 | \( 1 + (0.222 - 0.974i)T \) |
| 11 | \( 1 + (-0.433 - 0.900i)T \) |
| 13 | \( 1 + (0.900 - 0.433i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (0.974 - 0.222i)T \) |
| 23 | \( 1 + (-0.623 - 0.781i)T \) |
| 31 | \( 1 + (0.781 + 0.623i)T \) |
| 37 | \( 1 + (0.433 - 0.900i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (-0.781 + 0.623i)T \) |
| 47 | \( 1 + (0.433 + 0.900i)T \) |
| 53 | \( 1 + (0.623 - 0.781i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (0.974 + 0.222i)T \) |
| 67 | \( 1 + (-0.900 - 0.433i)T \) |
| 71 | \( 1 + (-0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.781 + 0.623i)T \) |
| 79 | \( 1 + (0.433 - 0.900i)T \) |
| 83 | \( 1 + (0.222 + 0.974i)T \) |
| 89 | \( 1 + (-0.781 - 0.623i)T \) |
| 97 | \( 1 + (0.974 - 0.222i)T \) |
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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
−29.01824941898829202422452067683, −28.07818720776756633995952198896, −28.0045138707455190384107846476, −26.54960962467090351063951244225, −25.24336249564625428785660519457, −24.04838891307437872733263205179, −23.44423769660564372720972554164, −22.31046411928928511153828055407, −21.26686962716827150937040760219, −20.37495862590621527361390166273, −18.8900401350727527210009535269, −17.96984457047643708535000233256, −16.88993231638279716452598757215, −15.754842611301208901268540526747, −15.25581512245171496679686967517, −13.27112158131978902765646268538, −12.12010044269049776262940447179, −11.62278603649217088238121689874, −10.15605098460293222276254798907, −8.905524712484351454484824047664, −7.64862781874604598457571140976, −6.03986714514650292274118354486, −5.05689157330948523732236474662, −3.90261087852374348219447745125, −1.58459555683099155808043533395,
0.805810677173947933354947396545, 3.154777090454597217788240345563, 4.548285138332564580475059931958, 6.00983141152939819668806516196, 7.1288491653420218380890036106, 8.07385315569728298062480758578, 10.1587955124936677048659224638, 11.01104858716008218420050187458, 11.7041185967274233381272300216, 13.24598166584979960846131968382, 14.177801419303273647848336124018, 15.85071560645917834148840029356, 16.37355240931701811954739917496, 17.91640020768339462710415832997, 18.42074813947235738483542082651, 19.69507044303859140345959330415, 20.926706137400270115005749795273, 22.236270854707221194350573771737, 23.0404848420067646156472046559, 23.70133859453373365311617676967, 24.76774954562044631175516407523, 26.46728311327306038954872528096, 26.94243738887097143122741667100, 28.05891814331039594922930064432, 29.181620380561271994229885985170