| L(s) = 1 | + 2·5-s − 4·7-s + 8·11-s − 6·13-s + 4·17-s − 8·19-s − 4·23-s − 3·25-s − 6·29-s + 4·31-s − 8·35-s − 4·37-s + 6·41-s − 12·47-s − 49-s − 4·53-s + 16·55-s + 8·59-s − 6·61-s − 12·65-s − 8·67-s − 8·71-s − 12·73-s − 32·77-s + 12·79-s − 8·83-s + 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 2.41·11-s − 1.66·13-s + 0.970·17-s − 1.83·19-s − 0.834·23-s − 3/5·25-s − 1.11·29-s + 0.718·31-s − 1.35·35-s − 0.657·37-s + 0.937·41-s − 1.75·47-s − 1/7·49-s − 0.549·53-s + 2.15·55-s + 1.04·59-s − 0.768·61-s − 1.48·65-s − 0.977·67-s − 0.949·71-s − 1.40·73-s − 3.64·77-s + 1.35·79-s − 0.878·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−13.4220897252, −13.2416026964, −12.5446568466, −12.5412064866, −11.8694020071, −11.7302167471, −11.0750661767, −10.3756594948, −9.87216608605, −9.82910081357, −9.42506276536, −9.03046954678, −8.48869306206, −7.83158898313, −7.24964163655, −6.70196192601, −6.44480626868, −5.95294980866, −5.67914881340, −4.61062307661, −4.28403219299, −3.56271769222, −3.08086895659, −2.12247216164, −1.61468837604, 0,
1.61468837604, 2.12247216164, 3.08086895659, 3.56271769222, 4.28403219299, 4.61062307661, 5.67914881340, 5.95294980866, 6.44480626868, 6.70196192601, 7.24964163655, 7.83158898313, 8.48869306206, 9.03046954678, 9.42506276536, 9.82910081357, 9.87216608605, 10.3756594948, 11.0750661767, 11.7302167471, 11.8694020071, 12.5412064866, 12.5446568466, 13.2416026964, 13.4220897252
