L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (0.669 − 0.743i)7-s + (−0.309 + 0.951i)8-s + 10-s + (0.913 − 0.406i)13-s + (−0.669 − 0.743i)14-s + (0.913 + 0.406i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)19-s + (0.104 − 0.994i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (0.669 − 0.743i)7-s + (−0.309 + 0.951i)8-s + 10-s + (0.913 − 0.406i)13-s + (−0.669 − 0.743i)14-s + (0.913 + 0.406i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)19-s + (0.104 − 0.994i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.323065686 - 1.220423221i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.323065686 - 1.220423221i\) |
\(L(1)\) |
\(\approx\) |
\(1.065776547 - 0.5726799819i\) |
\(L(1)\) |
\(\approx\) |
\(1.065776547 - 0.5726799819i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 7 | \( 1 + (0.669 - 0.743i)T \) |
| 13 | \( 1 + (0.913 - 0.406i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.669 + 0.743i)T \) |
| 31 | \( 1 + (0.913 - 0.406i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (-0.669 - 0.743i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.978 - 0.207i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.978 + 0.207i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.104 + 0.994i)T \) |
| 83 | \( 1 + (-0.913 - 0.406i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.104 + 0.994i)T \) |
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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
−30.275356090788808408326042105234, −28.50448589314261886546839868885, −27.90939253680868046978692893722, −26.85296125615109919698474168564, −25.46757976544743925250604127883, −24.874444461967917763174926870827, −23.87928373315259553699075139110, −23.042565271343559319020083806533, −21.495263245086381732669957072706, −20.91824367389411980394678579695, −19.13639254969798974264679220766, −18.08071096772885978075352131946, −17.02069469421266115734866285391, −16.11026188781268862759181253401, −15.06875139300860256356573337507, −13.91506941294883193135192714421, −12.809730293884851694386119921574, −11.70134585388793789174194785407, −9.71141732328793629233019435420, −8.61750443857073624759304810054, −7.85192781808624429255065453511, −6.03464953642237590756842339201, −5.211878790999980506062178506926, −3.86776509698060276246450437913, −1.3475337489348458251091627243,
0.96932535101209243553252746406, 2.67768686208666644446961854081, 3.87299905924880373818124037613, 5.347727107227446844293814107698, 7.07519175909121080112329368954, 8.49322878948863465373561065397, 10.04219283843087799042821252661, 10.86082853555645633715472327886, 11.72036297992842684331461093613, 13.34277692362880730814174832369, 14.095436694020189685996837725195, 15.184491347079391852633986407654, 17.0460727048521133586772151854, 18.14898731864724016956338216098, 18.82617358446832387763572666982, 20.215084012382348070939819647919, 20.93844030938481402659218585728, 22.15072316720653318266147568707, 22.99241583184029861193538695888, 23.90778863974263965598550574838, 25.637047457217972420804087992950, 26.68502555010572475599938010387, 27.43548533924638566447609519056, 28.56116989538361431457545709890, 29.834098044975568021064839871010