L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 6·12-s + 5·16-s − 6·18-s − 8·24-s + 4·27-s − 6·32-s + 9·36-s + 10·48-s + 2·49-s − 8·54-s + 7·64-s − 12·72-s + 5·81-s − 12·96-s − 4·97-s − 4·98-s + 12·108-s − 2·121-s + 127-s − 8·128-s + 131-s + ⋯ |
L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 6·12-s + 5·16-s − 6·18-s − 8·24-s + 4·27-s − 6·32-s + 9·36-s + 10·48-s + 2·49-s − 8·54-s + 7·64-s − 12·72-s + 5·81-s − 12·96-s − 4·97-s − 4·98-s + 12·108-s − 2·121-s + 127-s − 8·128-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.201126857\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.201126857\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.214066079282701444981799988734, −9.153185246128506526850677113033, −8.879276186849607427647040622757, −8.412961246670247356577413665962, −7.987965655892556815219143801371, −7.916895219853405224120416975132, −7.41051997880905346986112716278, −7.07082369257477852149992716431, −6.67083347587613773843148875436, −6.46815055191435980337744499833, −5.59971734959201756791184949995, −5.37846181979361614360849979197, −4.36522854736731233247351960732, −4.03619821377025191371761572377, −3.41593054739066540491244199593, −3.02775246984645232156406584006, −2.47821479992055549652802069702, −2.25298910163709615521970108812, −1.52787103337297115316766718154, −1.07868558643509318116617262004,
1.07868558643509318116617262004, 1.52787103337297115316766718154, 2.25298910163709615521970108812, 2.47821479992055549652802069702, 3.02775246984645232156406584006, 3.41593054739066540491244199593, 4.03619821377025191371761572377, 4.36522854736731233247351960732, 5.37846181979361614360849979197, 5.59971734959201756791184949995, 6.46815055191435980337744499833, 6.67083347587613773843148875436, 7.07082369257477852149992716431, 7.41051997880905346986112716278, 7.916895219853405224120416975132, 7.987965655892556815219143801371, 8.412961246670247356577413665962, 8.879276186849607427647040622757, 9.153185246128506526850677113033, 9.214066079282701444981799988734