| L(s) = 1 | + 61·4-s − 486·9-s + 2.69e3·16-s + 6.25e3·25-s − 2.96e4·36-s − 6.72e4·49-s − 1.39e5·61-s + 1.02e5·64-s + 1.77e5·81-s + 3.81e5·100-s − 5.03e5·109-s − 6.44e5·121-s + 127-s + 131-s + 137-s + 139-s − 1.31e6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.48e6·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.90·4-s − 2·9-s + 2.63·16-s + 2·25-s − 3.81·36-s − 4·49-s − 4.79·61-s + 3.11·64-s + 3·81-s + 3.81·100-s − 4.06·109-s − 4·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 5.26·144-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.722527368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.722527368\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - 61 T^{2} + p^{10} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 2419214 T^{2} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2164 T + p^{5} T^{2} )^{2}( 1 + 2164 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 10950686 T^{2} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8152 T + p^{5} T^{2} )^{2}( 1 + 8152 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 311808014 T^{2} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 836229514 T^{2} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 34802 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 70064 T + p^{5} T^{2} )^{2}( 1 + 70064 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 2642233286 T^{2} + p^{10} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.39711844001355369995742725265, −9.959543318701449538212728505803, −9.499625520159966222369892803579, −9.174704404545756585303668468812, −9.059110848667624021763671257759, −8.425147169944319396014685827529, −8.183638704468911770666832235231, −8.006212496260213293685912036165, −7.56601191716416551961177694114, −7.32911190645733895662670404555, −6.70599650658927333779178894389, −6.44202225634761342080298676913, −6.29756211969504577696191430782, −6.05030892001333180395095988349, −5.39390020442934926710204035211, −5.13291908889729115953599852162, −4.83569465323033961644590182682, −4.07811486877285194194074388872, −3.32551292687053011080103786044, −3.04464524414253636784218423785, −2.86431614034986513348992062963, −2.40896881552571179828165298562, −1.49171985602151090481865763173, −1.40547180846058450597192123906, −0.25821760885397899527095426079,
0.25821760885397899527095426079, 1.40547180846058450597192123906, 1.49171985602151090481865763173, 2.40896881552571179828165298562, 2.86431614034986513348992062963, 3.04464524414253636784218423785, 3.32551292687053011080103786044, 4.07811486877285194194074388872, 4.83569465323033961644590182682, 5.13291908889729115953599852162, 5.39390020442934926710204035211, 6.05030892001333180395095988349, 6.29756211969504577696191430782, 6.44202225634761342080298676913, 6.70599650658927333779178894389, 7.32911190645733895662670404555, 7.56601191716416551961177694114, 8.006212496260213293685912036165, 8.183638704468911770666832235231, 8.425147169944319396014685827529, 9.059110848667624021763671257759, 9.174704404545756585303668468812, 9.499625520159966222369892803579, 9.959543318701449538212728505803, 10.39711844001355369995742725265
