L(s) = 1 | + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.939 + 0.342i)5-s + (0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.173 + 0.984i)10-s + (−0.5 + 0.866i)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 + (0.173 − 0.984i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.939 + 0.342i)5-s + (0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.173 + 0.984i)10-s + (−0.5 + 0.866i)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 + (0.173 − 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6490920513 - 0.2418349987i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6490920513 - 0.2418349987i\) |
\(L(1)\) |
\(\approx\) |
\(0.9148777962 - 0.2743962000i\) |
\(L(1)\) |
\(\approx\) |
\(0.9148777962 - 0.2743962000i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.766 - 0.642i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.173 + 0.984i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (-0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.173 + 0.984i)T \) |
| 53 | \( 1 + (-0.939 - 0.342i)T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (-0.939 + 0.342i)T \) |
| 73 | \( 1 + (0.766 + 0.642i)T \) |
| 79 | \( 1 + (0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.173 - 0.984i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−41.20384797788594438920615324210, −39.75253373678277456012646920810, −37.92800505845793264133536334229, −36.33581719795455556071506717654, −35.3379868267910960637953750558, −34.48649083564432440906150833866, −32.348634645340665716708419904855, −31.6008207000210932899107105769, −30.57937888475565841125168579009, −28.38185295028579843677332387969, −26.683928658170236982917790133, −25.58986480145456883594150043559, −24.262187876552078686309678741301, −23.40258719388946822596410683783, −21.45392796599038916835575109704, −19.364707078271979697428408939988, −18.399608549824534518277132803751, −16.231741681749722530869331618, −15.15475284576543290999489815175, −13.5235662208293974187418359784, −12.219233258408520928330202026637, −8.919819492265205268438072345589, −7.96383872105274116227731779107, −6.144738739176663757367474316151, −3.63715141026986389406139395986,
3.098780796516242989029474866290, 4.40326503236999735608592608485, 7.87046245161347143463711975430, 9.784705571944299027829113931934, 10.97732081444656069231364161663, 12.961483732421802065524330534898, 14.45929740897332565709300613413, 15.97183270410677329366944613353, 18.42318407229489172366645497977, 19.95633891187922612740804620497, 20.500211947805143210805262350228, 22.37992520140482266628026460011, 23.38903181380005856896111069424, 25.90103103262920986888298480333, 27.01930258664812602884351785427, 28.0860176070842113378064833370, 29.97308117879326510240171457494, 31.02200993497157192939617987746, 32.074025094767684882108684988209, 33.3602343152603013879218081086, 35.65457837287952981220758062472, 36.69033350196856596357589369148, 38.25102936823066918362673207345, 38.79794059860006376489807231668, 39.966678468297088389924664394