L(s) = 1 | − 2·5-s + 2·7-s − 2·9-s + 4·13-s + 3·25-s − 8·31-s − 4·35-s − 20·43-s + 4·45-s − 12·47-s − 3·49-s + 4·61-s − 4·63-s − 8·65-s + 4·67-s − 5·81-s + 8·91-s + 12·101-s + 28·103-s − 12·107-s − 12·113-s − 8·117-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.755·7-s − 2/3·9-s + 1.10·13-s + 3/5·25-s − 1.43·31-s − 0.676·35-s − 3.04·43-s + 0.596·45-s − 1.75·47-s − 3/7·49-s + 0.512·61-s − 0.503·63-s − 0.992·65-s + 0.488·67-s − 5/9·81-s + 0.838·91-s + 1.19·101-s + 2.75·103-s − 1.16·107-s − 1.12·113-s − 0.739·117-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.552217550781204179646223792184, −8.274191454688856203377872859552, −7.67724524894114537171979145145, −7.31157654449653438132198826839, −6.57891116465648258947670054106, −6.36392101906843607751379404756, −5.60452081656466550382223684619, −5.07009055292225152793438727510, −4.78130792717525308450176413839, −3.95271496302851434324941322577, −3.49047115245004503734053909847, −3.10876796327263031806431254214, −2.02979685108826286576114913510, −1.36690026055352676588947736551, 0,
1.36690026055352676588947736551, 2.02979685108826286576114913510, 3.10876796327263031806431254214, 3.49047115245004503734053909847, 3.95271496302851434324941322577, 4.78130792717525308450176413839, 5.07009055292225152793438727510, 5.60452081656466550382223684619, 6.36392101906843607751379404756, 6.57891116465648258947670054106, 7.31157654449653438132198826839, 7.67724524894114537171979145145, 8.274191454688856203377872859552, 8.552217550781204179646223792184