| L(s) = 1 | − 4.09e3·4-s − 3.49e5·9-s + 1.25e7·16-s − 9.76e7·25-s + 1.42e9·36-s − 3.61e8·49-s + 1.90e10·61-s − 3.43e10·64-s + 9.04e10·81-s + 4.00e11·100-s − 2.27e11·109-s − 1.14e12·121-s + 127-s + 131-s + 137-s + 139-s − 4.39e12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.16e12·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·4-s − 1.97·9-s + 3·16-s − 2·25-s + 3.94·36-s − 0.182·49-s + 2.88·61-s − 4·64-s + 2.88·81-s + 4·100-s − 1.41·109-s − 4·121-s − 5.91·144-s + 4·169-s + 0.365·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.123266529\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.123266529\times10^{-5}\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 349004 T^{2} + p^{22} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 180570196 T^{2} + p^{22} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 1893250316982244 T^{2} + p^{22} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 47436306 T + p^{11} T^{2} )^{2}( 1 + 47436306 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1052185188 T + p^{11} T^{2} )^{2}( 1 + 1052185188 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 1117494535796148124 T^{2} + p^{22} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2231392497520513004 T^{2} + p^{22} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4754428808 T + p^{11} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - \)\(21\!\cdots\!84\)\( T^{2} + p^{22} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + \)\(22\!\cdots\!24\)\( T^{2} + p^{22} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 42789447894 T + p^{11} T^{2} )^{2}( 1 + 42789447894 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.651060468622875765377476778228, −8.535363443431198738075210417471, −8.405243395934296106311250903462, −7.929715019841015282764896132162, −7.63872855666710922085078262274, −7.60144195962546999634517747326, −6.68856865255624595954481456580, −6.58554050196209911862976390483, −6.06831250310613038223490492823, −5.65880918761348509234458712540, −5.55810113382123365480493592997, −5.24605972678037226647705979355, −4.96438282140228347326736474442, −4.48836708840896085275547256264, −3.97074045099285151972740270217, −3.85566391128951665164577849676, −3.56609265135124456253280341508, −3.04316042089314791233525108711, −2.71765699793797245348674556558, −2.26625722648501921394635646250, −1.84497050165400739374799199486, −1.23988827295281052846260402273, −0.876915812051207545274614012046, −0.45742927520424127513214257286, −0.00108844138643424017001685003,
0.00108844138643424017001685003, 0.45742927520424127513214257286, 0.876915812051207545274614012046, 1.23988827295281052846260402273, 1.84497050165400739374799199486, 2.26625722648501921394635646250, 2.71765699793797245348674556558, 3.04316042089314791233525108711, 3.56609265135124456253280341508, 3.85566391128951665164577849676, 3.97074045099285151972740270217, 4.48836708840896085275547256264, 4.96438282140228347326736474442, 5.24605972678037226647705979355, 5.55810113382123365480493592997, 5.65880918761348509234458712540, 6.06831250310613038223490492823, 6.58554050196209911862976390483, 6.68856865255624595954481456580, 7.60144195962546999634517747326, 7.63872855666710922085078262274, 7.929715019841015282764896132162, 8.405243395934296106311250903462, 8.535363443431198738075210417471, 8.651060468622875765377476778228
