| L(s) = 1 | + 2·3-s + 4-s + 9-s + 2·12-s − 5·13-s − 3·16-s + 12·23-s + 25-s − 4·27-s + 3·29-s + 36-s − 10·39-s − 2·43-s − 6·48-s − 13·49-s − 5·52-s − 12·53-s − 19·61-s − 7·64-s + 24·69-s + 2·75-s + 20·79-s − 11·81-s + 6·87-s + 12·92-s + 100-s + 4·103-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 1/3·9-s + 0.577·12-s − 1.38·13-s − 3/4·16-s + 2.50·23-s + 1/5·25-s − 0.769·27-s + 0.557·29-s + 1/6·36-s − 1.60·39-s − 0.304·43-s − 0.866·48-s − 1.85·49-s − 0.693·52-s − 1.64·53-s − 2.43·61-s − 7/8·64-s + 2.88·69-s + 0.230·75-s + 2.25·79-s − 1.22·81-s + 0.643·87-s + 1.25·92-s + 1/10·100-s + 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.549352471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549352471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 119 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.06497506627261472133487767077, −10.86741806008311573965484313662, −9.949944498665374559111795087246, −9.394622294783736933284183104670, −9.127375282336494428626329005588, −8.448072752662845043300421818243, −7.81228397533464418681895033489, −7.28510557614469024029326097876, −6.79665816181015761970055867954, −6.09536498296674918473920325711, −4.82499250153099788066604690002, −4.77467413176289562693031075711, −3.24019695075966564628351551676, −2.91660331072002567980662448844, −1.93366997904206856491583206168,
1.93366997904206856491583206168, 2.91660331072002567980662448844, 3.24019695075966564628351551676, 4.77467413176289562693031075711, 4.82499250153099788066604690002, 6.09536498296674918473920325711, 6.79665816181015761970055867954, 7.28510557614469024029326097876, 7.81228397533464418681895033489, 8.448072752662845043300421818243, 9.127375282336494428626329005588, 9.394622294783736933284183104670, 9.949944498665374559111795087246, 10.86741806008311573965484313662, 11.06497506627261472133487767077
