L(s) = 1 | + (−0.997 + 0.0729i)2-s + (−0.181 + 0.983i)3-s + (0.989 − 0.145i)4-s + (0.639 − 0.768i)5-s + (0.109 − 0.994i)6-s + (−0.694 + 0.719i)7-s + (−0.976 + 0.217i)8-s + (−0.934 − 0.357i)9-s + (−0.581 + 0.813i)10-s + (0.744 − 0.667i)11-s + (−0.0365 + 0.999i)12-s + (0.109 + 0.994i)13-s + (0.639 − 0.768i)14-s + (0.639 + 0.768i)15-s + (0.957 − 0.288i)16-s + (−0.872 + 0.489i)17-s + ⋯ |
L(s) = 1 | + (−0.997 + 0.0729i)2-s + (−0.181 + 0.983i)3-s + (0.989 − 0.145i)4-s + (0.639 − 0.768i)5-s + (0.109 − 0.994i)6-s + (−0.694 + 0.719i)7-s + (−0.976 + 0.217i)8-s + (−0.934 − 0.357i)9-s + (−0.581 + 0.813i)10-s + (0.744 − 0.667i)11-s + (−0.0365 + 0.999i)12-s + (0.109 + 0.994i)13-s + (0.639 − 0.768i)14-s + (0.639 + 0.768i)15-s + (0.957 − 0.288i)16-s + (−0.872 + 0.489i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3164978619 - 0.2778189325i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3164978619 - 0.2778189325i\) |
\(L(1)\) |
\(\approx\) |
\(0.5857033614 + 0.09430219268i\) |
\(L(1)\) |
\(\approx\) |
\(0.5857033614 + 0.09430219268i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (-0.997 + 0.0729i)T \) |
| 3 | \( 1 + (-0.181 + 0.983i)T \) |
| 5 | \( 1 + (0.639 - 0.768i)T \) |
| 7 | \( 1 + (-0.694 + 0.719i)T \) |
| 11 | \( 1 + (0.744 - 0.667i)T \) |
| 13 | \( 1 + (0.109 + 0.994i)T \) |
| 17 | \( 1 + (-0.872 + 0.489i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.0365 - 0.999i)T \) |
| 29 | \( 1 + (-0.934 + 0.357i)T \) |
| 31 | \( 1 + (0.109 - 0.994i)T \) |
| 37 | \( 1 + (0.252 - 0.967i)T \) |
| 41 | \( 1 + (-0.934 - 0.357i)T \) |
| 47 | \( 1 + (-0.934 + 0.357i)T \) |
| 53 | \( 1 + (-0.976 - 0.217i)T \) |
| 59 | \( 1 + (0.905 + 0.424i)T \) |
| 61 | \( 1 + (-0.694 + 0.719i)T \) |
| 67 | \( 1 + (-0.997 + 0.0729i)T \) |
| 71 | \( 1 + (-0.934 - 0.357i)T \) |
| 73 | \( 1 + (0.989 + 0.145i)T \) |
| 79 | \( 1 + (-0.322 - 0.946i)T \) |
| 83 | \( 1 + (-0.694 + 0.719i)T \) |
| 89 | \( 1 + (0.391 + 0.920i)T \) |
| 97 | \( 1 + (0.109 - 0.994i)T \) |
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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
−20.015276624009057647530675899829, −19.578653694183259228679582300193, −18.64091566972805776175449723478, −18.11566712919760255495425417620, −17.40701337783344519120639753617, −17.14883911699563621850274161435, −16.066710545741762753638393680190, −15.24594026288491684157631594486, −14.35579686699472517746451397054, −13.461034811379226273274953641874, −12.974424543452415383142546304074, −11.88033202942755568510892299886, −11.31895807623208783401682920494, −10.47422192755215988360910950227, −9.7544948724184143396976464734, −9.17350521231317784070927041833, −7.96593789812391619138683882010, −7.29973732058618707593037063647, −6.75307008091896542655746428033, −6.20542990532149699537714468089, −5.21626544969730756185915751658, −3.42521406680864053280448240213, −2.92332106183901470912135280383, −1.81816539488635956795353899778, −1.15358131796618116771475702752,
0.21694473481030267742044711068, 1.56799773261170835102303150719, 2.49095368763495044195412561555, 3.50564244235837193778822678971, 4.48516348155552355257401895300, 5.64091806217617208632260560555, 6.08537588498640442442971813423, 6.83396374742061525095414933945, 8.31257137500939644852145370039, 8.95176064281333833936028106781, 9.30004017764765808543292369838, 9.89303099722750955527147510221, 10.88007521882447398792138312143, 11.59191166273537621878237340285, 12.216732233895013000396337689390, 13.271866256875121773110756947678, 14.29251245988180825431840144557, 15.02920933767944497864818466346, 15.93433925274012314481131818062, 16.468799331534692768684404729891, 16.76546916471138734020650597974, 17.66689309567001115913594435083, 18.41369030599844013679095213453, 19.295492118318321475181988006532, 19.92487715351907746762392047716