L(s) = 1 | + 2·5-s − 4·7-s − 3·9-s − 12·17-s + 3·25-s − 8·35-s − 4·37-s − 12·41-s + 16·43-s − 6·45-s − 24·47-s + 9·49-s − 24·59-s + 12·63-s − 16·67-s − 16·79-s + 9·81-s − 24·85-s − 12·89-s + 12·101-s − 4·109-s + 48·119-s + 10·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.51·7-s − 9-s − 2.91·17-s + 3/5·25-s − 1.35·35-s − 0.657·37-s − 1.87·41-s + 2.43·43-s − 0.894·45-s − 3.50·47-s + 9/7·49-s − 3.12·59-s + 1.51·63-s − 1.95·67-s − 1.80·79-s + 81-s − 2.60·85-s − 1.27·89-s + 1.19·101-s − 0.383·109-s + 4.40·119-s + 0.909·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.316073854395624004154864942662, −8.940568158013617805848144359499, −8.462506867449611836284672627400, −8.162676340682739864153128541061, −7.44957541891435189386794357432, −6.98406684796553792642373238201, −6.53406125129806104439694310132, −6.49278714686704891141662457183, −5.89566410038900288749051494653, −5.77310515564915704847389379378, −4.97152381819404007914555175910, −4.60109187036206067787408291834, −4.25044293194855338737565881549, −3.35928359512102330716724073897, −3.10149280509513910286441803020, −2.67574019294977927566316448883, −2.03803088670710521253178010633, −1.55679575480608851551863570636, 0, 0,
1.55679575480608851551863570636, 2.03803088670710521253178010633, 2.67574019294977927566316448883, 3.10149280509513910286441803020, 3.35928359512102330716724073897, 4.25044293194855338737565881549, 4.60109187036206067787408291834, 4.97152381819404007914555175910, 5.77310515564915704847389379378, 5.89566410038900288749051494653, 6.49278714686704891141662457183, 6.53406125129806104439694310132, 6.98406684796553792642373238201, 7.44957541891435189386794357432, 8.162676340682739864153128541061, 8.462506867449611836284672627400, 8.940568158013617805848144359499, 9.316073854395624004154864942662