L(s) = 1 | + 5-s − 2·7-s + 11-s − 13-s + 2·17-s + 8·19-s + 23-s − 5·29-s − 31-s − 2·35-s + 12·37-s − 7·43-s + 7·47-s + 7·49-s + 24·53-s + 55-s − 4·59-s − 10·61-s − 65-s + 4·67-s − 24·71-s + 12·73-s − 2·77-s − 15·79-s + 2·83-s + 2·85-s + 24·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.301·11-s − 0.277·13-s + 0.485·17-s + 1.83·19-s + 0.208·23-s − 0.928·29-s − 0.179·31-s − 0.338·35-s + 1.97·37-s − 1.06·43-s + 1.02·47-s + 49-s + 3.29·53-s + 0.134·55-s − 0.520·59-s − 1.28·61-s − 0.124·65-s + 0.488·67-s − 2.84·71-s + 1.40·73-s − 0.227·77-s − 1.68·79-s + 0.219·83-s + 0.216·85-s + 2.54·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.741263987\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.741263987\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.978996037818273424428764250109, −8.540223972768996793226948795706, −8.030931578743356619696315012254, −7.59762911680978690345969509947, −7.31804891056379523800901404089, −7.13283953037680506822708484828, −6.55536425793309974024182979886, −6.19879515558044651203530969823, −5.71957074898778087340546010500, −5.47748566771949301448685266056, −5.25117483585517765213115052153, −4.55833196703981417968026599152, −4.03304808304314484724377918909, −3.85022534545496910170789524940, −3.13818904541381699451621627970, −2.86739079588172408049261016459, −2.44945930128364634824539687876, −1.71498687479888353872801320952, −1.14665309631874332067934706220, −0.56611576397386358411542041566,
0.56611576397386358411542041566, 1.14665309631874332067934706220, 1.71498687479888353872801320952, 2.44945930128364634824539687876, 2.86739079588172408049261016459, 3.13818904541381699451621627970, 3.85022534545496910170789524940, 4.03304808304314484724377918909, 4.55833196703981417968026599152, 5.25117483585517765213115052153, 5.47748566771949301448685266056, 5.71957074898778087340546010500, 6.19879515558044651203530969823, 6.55536425793309974024182979886, 7.13283953037680506822708484828, 7.31804891056379523800901404089, 7.59762911680978690345969509947, 8.030931578743356619696315012254, 8.540223972768996793226948795706, 8.978996037818273424428764250109