L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s − 13-s + 15-s + 17-s + 19-s + 21-s + 23-s + 25-s − 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + ⋯ |
L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s − 13-s + 15-s + 17-s + 19-s + 21-s + 23-s + 25-s − 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2348 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2348 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7087340314\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7087340314\) |
\(L(1)\) |
\(\approx\) |
\(0.6254838274\) |
\(L(1)\) |
\(\approx\) |
\(0.6254838274\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 587 | \( 1 \) |
good | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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
−19.67333724292221522512104597643, −18.823271487742174420767963787949, −18.32105591094818071720742893996, −17.146177800772302876276569876642, −16.743297322115605415930555354776, −16.18642374634967463410625971338, −15.401351577817466511996137763329, −14.762108750652480147983471892516, −13.77548288215325778563294563199, −12.774117679785958328611294948446, −12.08229114941215463140063198330, −11.90492411289250037056081464657, −10.948328795762844351094799278279, −10.07381147378783789382134582730, −9.5357292479733762146963005813, −8.58382957226941129612702433225, −7.33373686458774408774160975251, −7.09173452195590922371763398989, −6.25858657270666129251843534880, −5.25321992522607798208751159344, −4.65757456198944199051348880795, −3.56484144730225331734199743982, −3.13908611128303646846942648589, −1.51198992553563685506780998969, −0.56077036460322639583224562614,
0.56077036460322639583224562614, 1.51198992553563685506780998969, 3.13908611128303646846942648589, 3.56484144730225331734199743982, 4.65757456198944199051348880795, 5.25321992522607798208751159344, 6.25858657270666129251843534880, 7.09173452195590922371763398989, 7.33373686458774408774160975251, 8.58382957226941129612702433225, 9.5357292479733762146963005813, 10.07381147378783789382134582730, 10.948328795762844351094799278279, 11.90492411289250037056081464657, 12.08229114941215463140063198330, 12.774117679785958328611294948446, 13.77548288215325778563294563199, 14.762108750652480147983471892516, 15.401351577817466511996137763329, 16.18642374634967463410625971338, 16.743297322115605415930555354776, 17.146177800772302876276569876642, 18.32105591094818071720742893996, 18.823271487742174420767963787949, 19.67333724292221522512104597643