L(s) = 1 | + 4·11-s + 4·23-s − 4·29-s − 4·37-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
L(s) = 1 | + 4·11-s + 4·23-s − 4·29-s − 4·37-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.334655283\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.334655283\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.41308578932996328607184793168, −5.89318797626734875488982857651, −5.82334399049606068110052591746, −5.66895450037369079482594448878, −5.54905046090371344498692225880, −5.26601461054313002469683555138, −4.87983044166724496821516152477, −4.85305527769848657169341325760, −4.84090532637585864881173315109, −4.26690851059388906974143169980, −4.19410894358792032749728721280, −3.95926916136639188772365098370, −3.74922575953610651330772104277, −3.52202474950726525188352021695, −3.38597114131175002917153142418, −3.23205804989832047987685988740, −3.14739466895736509218870991284, −2.70305810637319658797469385740, −2.09369277184077846606514369400, −2.05716030755713228218105476889, −1.90039386967745192039866151846, −1.33818556008429459999772826554, −1.31980013440966420702442776858, −1.24491026419306281339092878023, −0.57184627605194205377383849911,
0.57184627605194205377383849911, 1.24491026419306281339092878023, 1.31980013440966420702442776858, 1.33818556008429459999772826554, 1.90039386967745192039866151846, 2.05716030755713228218105476889, 2.09369277184077846606514369400, 2.70305810637319658797469385740, 3.14739466895736509218870991284, 3.23205804989832047987685988740, 3.38597114131175002917153142418, 3.52202474950726525188352021695, 3.74922575953610651330772104277, 3.95926916136639188772365098370, 4.19410894358792032749728721280, 4.26690851059388906974143169980, 4.84090532637585864881173315109, 4.85305527769848657169341325760, 4.87983044166724496821516152477, 5.26601461054313002469683555138, 5.54905046090371344498692225880, 5.66895450037369079482594448878, 5.82334399049606068110052591746, 5.89318797626734875488982857651, 6.41308578932996328607184793168