| L(s) = 1 | + 3·2-s + 3·4-s + 7-s − 3·8-s + 2·9-s + 4·11-s + 3·14-s − 13·16-s + 6·18-s + 12·22-s − 4·23-s + 4·25-s + 3·28-s − 4·29-s − 15·32-s + 6·36-s − 19·37-s − 7·43-s + 12·44-s − 12·46-s − 6·49-s + 12·50-s + 5·53-s − 3·56-s − 12·58-s + 2·63-s + 3·64-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s + 0.377·7-s − 1.06·8-s + 2/3·9-s + 1.20·11-s + 0.801·14-s − 3.25·16-s + 1.41·18-s + 2.55·22-s − 0.834·23-s + 4/5·25-s + 0.566·28-s − 0.742·29-s − 2.65·32-s + 36-s − 3.12·37-s − 1.06·43-s + 1.80·44-s − 1.76·46-s − 6/7·49-s + 1.69·50-s + 0.686·53-s − 0.400·56-s − 1.57·58-s + 0.251·63-s + 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1645567 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1645567 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 12 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
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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
−7.30632058494599380054055056279, −7.00315789921653367328950246187, −6.58106055746413113056851328016, −6.29354323658527569444343329854, −5.66471517363655632918702801672, −5.24138493336419765024064280046, −5.06492396909371207355675797649, −4.45739408449607958944645512336, −4.08255965397155433719543814974, −3.70724092088203652966821915887, −3.38763383724293672560795593317, −2.76521804356587140189335911171, −1.92099673422861797016111438841, −1.40402873982953243427890378198, 0,
1.40402873982953243427890378198, 1.92099673422861797016111438841, 2.76521804356587140189335911171, 3.38763383724293672560795593317, 3.70724092088203652966821915887, 4.08255965397155433719543814974, 4.45739408449607958944645512336, 5.06492396909371207355675797649, 5.24138493336419765024064280046, 5.66471517363655632918702801672, 6.29354323658527569444343329854, 6.58106055746413113056851328016, 7.00315789921653367328950246187, 7.30632058494599380054055056279
