L(s) = 1 | + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.5 − 0.866i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.766 − 0.642i)10-s + 11-s + (−0.5 + 0.866i)12-s + (0.766 − 0.642i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.5 − 0.866i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.766 − 0.642i)10-s + 11-s + (−0.5 + 0.866i)12-s + (0.766 − 0.642i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.438161457 + 0.2011101971i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.438161457 + 0.2011101971i\) |
\(L(1)\) |
\(\approx\) |
\(1.281825395 + 0.2305965121i\) |
\(L(1)\) |
\(\approx\) |
\(1.281825395 + 0.2305965121i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (0.766 - 0.642i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.173 + 0.984i)T \) |
| 31 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (0.173 - 0.984i)T \) |
| 41 | \( 1 + (-0.939 + 0.342i)T \) |
| 43 | \( 1 + (0.173 + 0.984i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (-0.939 + 0.342i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.939 + 0.342i)T \) |
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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
−27.0405280556152225858266981585, −26.39710940117566046282937694328, −25.186884706672083340222761218236, −23.72215368647704735533149202034, −22.77065301902154522028398420920, −22.122621028832599667716012179607, −20.97354386196931080553358761401, −20.279367443794342708428594767692, −19.42534266358782505019134741074, −18.71853783991524935859534097222, −17.43385412853344007414679257559, −16.1100606527365079913635152721, −14.92334244858133389875871065598, −13.961656591060321190912661351081, −13.6340889983759281840001810480, −11.752951193410170743696740213398, −11.09345709217750695959130430023, −10.10993163979889636987356308942, −9.21713460751592710165104420026, −8.02071820234363262988873906875, −6.73328748316112464979479016975, −4.76682302914839037436352025572, −3.79281557398078028626881477304, −3.174257207372100303199189023349, −1.574704485024391576362103856241,
1.2406198302051021516739330862, 3.18987709224218661213701755320, 4.38081969377021132619974542405, 5.71434526281222654478909209636, 6.77447479611848746971034061160, 8.037184505829370667218540000067, 8.715136756735640702041696106776, 9.316382867349338032897886879113, 11.6033328271465348774672404420, 12.67858760875350521611524112036, 13.22251006035641658403873533965, 14.55809262077386397519727791667, 15.23858208064577096176702795069, 16.13119199168263468078874883392, 17.375714776886948357416229651017, 18.23212164593841697183002714969, 19.2383183902651341721688847525, 20.176803203900094970907990955854, 21.319828889940260813712337963014, 22.44188856419113045739202940809, 23.55250762404045230875014097227, 24.47368785027773511548321475465, 24.83699772255159344107785437316, 25.6943701698996934307371812858, 26.77074335804570313360746293781