L(s) = 1 | + (−0.913 + 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)5-s + (−0.978 − 0.207i)7-s + (−0.309 + 0.951i)8-s + 10-s + (−0.104 + 0.994i)13-s + (0.978 − 0.207i)14-s + (−0.104 − 0.994i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
L(s) = 1 | + (−0.913 + 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)5-s + (−0.978 − 0.207i)7-s + (−0.309 + 0.951i)8-s + 10-s + (−0.104 + 0.994i)13-s + (0.978 − 0.207i)14-s + (−0.104 − 0.994i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6122694690 + 0.3161635852i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6122694690 + 0.3161635852i\) |
\(L(1)\) |
\(\approx\) |
\(0.5768824020 + 0.09460237375i\) |
\(L(1)\) |
\(\approx\) |
\(0.5768824020 + 0.09460237375i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.913 + 0.406i)T \) |
| 5 | \( 1 + (-0.913 - 0.406i)T \) |
| 7 | \( 1 + (-0.978 - 0.207i)T \) |
| 13 | \( 1 + (-0.104 + 0.994i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.978 + 0.207i)T \) |
| 31 | \( 1 + (-0.104 + 0.994i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.978 - 0.207i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.669 - 0.743i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.669 + 0.743i)T \) |
| 61 | \( 1 + (-0.104 - 0.994i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.104 + 0.994i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.913 - 0.406i)T \) |
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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
−29.5127036281674695463858343524, −28.42505127470613203419887422353, −27.50153548723594445216267135991, −26.6408451428212735007097545113, −25.677720883907303287774536560468, −24.717526038840909336452563995870, −23.107397136854045605558265996848, −22.29895827379357775200522092185, −20.89977305915120792350317724960, −19.77966217199194654968281988540, −19.072424267549644876612832166017, −18.183447820228128343244090130445, −16.742080356119349810866235800117, −15.89050509070542439265432169695, −14.827717822704560556464086640109, −12.80515240793507262096448812352, −12.07349601066620018342649692192, −10.71468243711158070126181842250, −9.85528463304591693740603278355, −8.39177857295027126561127567921, −7.45525306465736327017619772172, −6.12862035130248689487175697490, −3.763014758900906442338302924418, −2.76617356100218558015762897176, −0.56496239421396663821170223177,
0.9455892516800216984267882008, 3.11987353217343080422767450243, 4.93131162947237142854119982078, 6.63055851144842764317919089834, 7.51063117126112503214776643961, 8.87440631621142962700186093735, 9.75588387752761666343846998404, 11.21004093189309122037682446186, 12.22317332660813454333170896808, 13.83414220545563221396393352969, 15.29921083044685699484884347577, 16.17388276700198196172522278545, 16.85273390103764884416123532747, 18.32386412299686046979109826239, 19.435311914381555070400078920282, 19.864060675259481062262133910926, 21.30483475840892450759864474406, 23.032880569482877794144015393089, 23.69722678501548867630301524014, 24.84665490428344321846439069973, 25.9207348117363486271846533295, 26.784264142132990522595479355829, 27.72607924383962655571350418955, 28.704297747916902309324049952309, 29.50384083836446040412934575129