| L(s) = 1 | − 2·3-s + 9-s − 4·11-s + 4·23-s + 25-s + 4·27-s − 8·31-s + 8·33-s − 12·37-s − 12·47-s − 2·49-s + 16·53-s + 16·67-s − 8·69-s − 8·71-s − 2·75-s − 11·81-s + 32·89-s + 16·93-s − 20·97-s − 4·99-s + 8·103-s + 24·111-s + 16·113-s + 5·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.20·11-s + 0.834·23-s + 1/5·25-s + 0.769·27-s − 1.43·31-s + 1.39·33-s − 1.97·37-s − 1.75·47-s − 2/7·49-s + 2.19·53-s + 1.95·67-s − 0.963·69-s − 0.949·71-s − 0.230·75-s − 1.22·81-s + 3.39·89-s + 1.65·93-s − 2.03·97-s − 0.402·99-s + 0.788·103-s + 2.27·111-s + 1.50·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−7.39872810031446104192797451649, −7.20699802287065575414472274264, −6.72668855065703536697699281084, −6.34231669840788912751245696184, −5.76008893389257096172329418778, −5.39340902188167415120250144953, −5.03386791287269834941254017126, −4.87891644486255913534883180248, −4.06964714631677540955167342584, −3.46149936145071072320863582123, −3.08896901022385549874361602073, −2.31789727479022948817187074178, −1.77050652143092979040469435264, −0.796544918596229829390449759573, 0,
0.796544918596229829390449759573, 1.77050652143092979040469435264, 2.31789727479022948817187074178, 3.08896901022385549874361602073, 3.46149936145071072320863582123, 4.06964714631677540955167342584, 4.87891644486255913534883180248, 5.03386791287269834941254017126, 5.39340902188167415120250144953, 5.76008893389257096172329418778, 6.34231669840788912751245696184, 6.72668855065703536697699281084, 7.20699802287065575414472274264, 7.39872810031446104192797451649
