| L(s) = 1 | + 2-s + 2·3-s − 4-s + 2·6-s − 3·8-s + 3·9-s − 2·12-s − 16-s + 3·18-s − 6·24-s + 6·25-s + 4·27-s + 5·32-s − 3·36-s + 12·41-s − 2·48-s + 2·49-s + 6·50-s + 4·54-s − 8·59-s + 7·64-s − 9·72-s + 12·73-s + 12·75-s + 5·81-s + 12·82-s + 10·96-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.816·6-s − 1.06·8-s + 9-s − 0.577·12-s − 1/4·16-s + 0.707·18-s − 1.22·24-s + 6/5·25-s + 0.769·27-s + 0.883·32-s − 1/2·36-s + 1.87·41-s − 0.288·48-s + 2/7·49-s + 0.848·50-s + 0.544·54-s − 1.04·59-s + 7/8·64-s − 1.06·72-s + 1.40·73-s + 1.38·75-s + 5/9·81-s + 1.32·82-s + 1.02·96-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.247551087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.247551087\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.920790519114965671257010703910, −8.454348571562701433984302858901, −7.85392252500592886735780157959, −7.60397505367903767130862241085, −6.92478978405668928717286262994, −6.42863197942740220204602984579, −5.94699188325599470659884992821, −5.25903998055197925567753027283, −4.85154726517137833091566815759, −4.20883043135713897973718086472, −3.91624558429943130112244988038, −3.12205759743206158519825507723, −2.83074343481662872937191278964, −2.04919469679826535599177793432, −0.926306969570920423508871513209,
0.926306969570920423508871513209, 2.04919469679826535599177793432, 2.83074343481662872937191278964, 3.12205759743206158519825507723, 3.91624558429943130112244988038, 4.20883043135713897973718086472, 4.85154726517137833091566815759, 5.25903998055197925567753027283, 5.94699188325599470659884992821, 6.42863197942740220204602984579, 6.92478978405668928717286262994, 7.60397505367903767130862241085, 7.85392252500592886735780157959, 8.454348571562701433984302858901, 8.920790519114965671257010703910
