| L(s) = 1 | + 2·3-s − 4-s + 2·5-s + 4·7-s + 9-s − 2·12-s + 4·15-s + 16-s − 2·20-s + 8·21-s − 25-s − 4·27-s − 4·28-s + 8·35-s − 36-s − 8·37-s − 16·41-s + 2·45-s − 8·47-s + 2·48-s + 9·49-s + 8·59-s − 4·60-s + 4·63-s − 64-s − 2·75-s + 24·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.577·12-s + 1.03·15-s + 1/4·16-s − 0.447·20-s + 1.74·21-s − 1/5·25-s − 0.769·27-s − 0.755·28-s + 1.35·35-s − 1/6·36-s − 1.31·37-s − 2.49·41-s + 0.298·45-s − 1.16·47-s + 0.288·48-s + 9/7·49-s + 1.04·59-s − 0.516·60-s + 0.503·63-s − 1/8·64-s − 0.230·75-s + 2.70·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.240630764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.240630764\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
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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
−10.04718597311099490382941178713, −9.651851440753352792144120936201, −9.062264716802331191966290208898, −8.608140191480052736692173723051, −8.203893505401840659724395040343, −7.913676130004262889961627520504, −7.11467954058939726069347084407, −6.55690630851794441069962721507, −5.63323247570334022739633833883, −5.21021770322333527848222007853, −4.69208066693068389012216132833, −3.81323272004505134172103727638, −3.19482895266048821376741662951, −2.11145581247912877553009275183, −1.67060615440643566270822936347,
1.67060615440643566270822936347, 2.11145581247912877553009275183, 3.19482895266048821376741662951, 3.81323272004505134172103727638, 4.69208066693068389012216132833, 5.21021770322333527848222007853, 5.63323247570334022739633833883, 6.55690630851794441069962721507, 7.11467954058939726069347084407, 7.913676130004262889961627520504, 8.203893505401840659724395040343, 8.608140191480052736692173723051, 9.062264716802331191966290208898, 9.651851440753352792144120936201, 10.04718597311099490382941178713
