| L(s) = 1 | − 2·3-s − 3·9-s − 8·11-s − 4·17-s + 12·19-s + 6·25-s + 14·27-s + 16·33-s + 6·41-s + 16·43-s − 10·49-s + 8·51-s − 24·57-s − 8·59-s − 8·67-s − 30·73-s − 12·75-s − 4·81-s + 12·83-s − 16·89-s + 20·97-s + 24·99-s + 12·107-s + 12·113-s + 26·121-s − 12·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 9-s − 2.41·11-s − 0.970·17-s + 2.75·19-s + 6/5·25-s + 2.69·27-s + 2.78·33-s + 0.937·41-s + 2.43·43-s − 1.42·49-s + 1.12·51-s − 3.17·57-s − 1.04·59-s − 0.977·67-s − 3.51·73-s − 1.38·75-s − 4/9·81-s + 1.31·83-s − 1.69·89-s + 2.03·97-s + 2.41·99-s + 1.16·107-s + 1.12·113-s + 2.36·121-s − 1.08·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270848 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270848 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.910132789738256271513641474783, −7.968671881522958272633798639034, −7.74681990799276192851943075763, −7.33683533600380364349770730667, −6.72708918808602349214044214496, −5.92862306306222724555843061273, −5.76108536627428988614940167857, −5.38734001699377202464167768623, −4.76193509639702859263858254048, −4.60774849630753565059580112097, −3.15021040115227360476019515636, −2.99122119577828822953767314363, −2.43771033806504815294329797529, −0.973947938192528970065157443625, 0,
0.973947938192528970065157443625, 2.43771033806504815294329797529, 2.99122119577828822953767314363, 3.15021040115227360476019515636, 4.60774849630753565059580112097, 4.76193509639702859263858254048, 5.38734001699377202464167768623, 5.76108536627428988614940167857, 5.92862306306222724555843061273, 6.72708918808602349214044214496, 7.33683533600380364349770730667, 7.74681990799276192851943075763, 7.968671881522958272633798639034, 8.910132789738256271513641474783
