L(s) = 1 | − 2.23e4·5-s + 8.90e6·13-s − 7.25e7·17-s + 1.21e8·25-s + 7.28e8·29-s − 2.72e9·37-s + 7.16e9·41-s + 4.73e10·49-s − 3.26e10·53-s − 1.30e11·61-s − 1.99e11·65-s − 1.70e11·73-s + 1.62e12·85-s − 2.01e12·89-s + 1.49e12·97-s + 1.78e11·101-s − 3.70e12·109-s − 1.95e12·113-s + 2.24e11·121-s − 2.52e12·125-s + 127-s + 131-s + 137-s + 139-s − 1.63e13·145-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 1.43·5-s + 1.84·13-s − 3.00·17-s + 0.497·25-s + 1.22·29-s − 1.06·37-s + 1.50·41-s + 3.42·49-s − 1.47·53-s − 2.53·61-s − 2.64·65-s − 1.12·73-s + 4.30·85-s − 4.04·89-s + 1.79·97-s + 0.168·101-s − 2.20·109-s − 0.939·113-s + 0.0716·121-s − 0.663·125-s − 1.75·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(0.01429145942\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01429145942\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.29633040232392443647898312721, −7.05271043542396182510326989955, −7.04718812981834925136541328875, −6.26135546824912030744458347174, −6.19522702714411441391222837820, −6.16772072010799184949257132151, −5.94065294069143562779693229466, −5.24611130516681982985649177949, −4.93388577174635670151004897815, −4.80656828556202259166869539988, −4.38323443342695217987462706248, −4.18716403479464095145586898881, −3.85760362906002473293337533071, −3.66703146793398472491953910968, −3.66519028689400111530591845244, −2.88772313817143026291890551822, −2.62350767515982957531964988927, −2.42316815838465671857241637497, −2.35915648404895311092608798776, −1.46867961831364473836248358598, −1.42444840865528461135039951281, −1.15268029725002265520136437166, −0.927762164319832476666237228757, −0.14075679068123061089199330297, −0.04252827997148282842156729805,
0.04252827997148282842156729805, 0.14075679068123061089199330297, 0.927762164319832476666237228757, 1.15268029725002265520136437166, 1.42444840865528461135039951281, 1.46867961831364473836248358598, 2.35915648404895311092608798776, 2.42316815838465671857241637497, 2.62350767515982957531964988927, 2.88772313817143026291890551822, 3.66519028689400111530591845244, 3.66703146793398472491953910968, 3.85760362906002473293337533071, 4.18716403479464095145586898881, 4.38323443342695217987462706248, 4.80656828556202259166869539988, 4.93388577174635670151004897815, 5.24611130516681982985649177949, 5.94065294069143562779693229466, 6.16772072010799184949257132151, 6.19522702714411441391222837820, 6.26135546824912030744458347174, 7.04718812981834925136541328875, 7.05271043542396182510326989955, 7.29633040232392443647898312721