L(s) = 1 | − 3-s + 9-s − 4·13-s − 8·23-s + 25-s − 27-s + 12·37-s + 4·39-s + 8·47-s + 2·49-s − 16·59-s − 4·61-s + 8·69-s − 16·71-s − 12·73-s − 75-s + 81-s − 8·83-s + 4·97-s + 8·107-s − 4·109-s − 12·111-s − 4·117-s − 6·121-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.10·13-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.97·37-s + 0.640·39-s + 1.16·47-s + 2/7·49-s − 2.08·59-s − 0.512·61-s + 0.963·69-s − 1.89·71-s − 1.40·73-s − 0.115·75-s + 1/9·81-s − 0.878·83-s + 0.406·97-s + 0.773·107-s − 0.383·109-s − 1.13·111-s − 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.078708301627855748942412226804, −8.461752277072843178252316260334, −7.79769161086611924559008207932, −7.53499221326312945395245645011, −7.13396987341472701822039150570, −6.28338669362981148011652990396, −6.04626156099497938177832826575, −5.57240268932539778004626159915, −4.77595227894881490913735587224, −4.43339993535207500187906625729, −3.89220098338467170975469010739, −2.91529246872248394447604670713, −2.36500969691943004667489432794, −1.38799300699998101566047980898, 0,
1.38799300699998101566047980898, 2.36500969691943004667489432794, 2.91529246872248394447604670713, 3.89220098338467170975469010739, 4.43339993535207500187906625729, 4.77595227894881490913735587224, 5.57240268932539778004626159915, 6.04626156099497938177832826575, 6.28338669362981148011652990396, 7.13396987341472701822039150570, 7.53499221326312945395245645011, 7.79769161086611924559008207932, 8.461752277072843178252316260334, 9.078708301627855748942412226804