| L(s) = 1 | + 9-s − 2·11-s + 6·19-s + 12·29-s − 8·31-s + 4·41-s + 6·49-s − 4·59-s − 8·61-s + 14·71-s − 2·79-s + 81-s + 20·89-s − 2·99-s + 8·101-s + 16·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 0.603·11-s + 1.37·19-s + 2.22·29-s − 1.43·31-s + 0.624·41-s + 6/7·49-s − 0.520·59-s − 1.02·61-s + 1.66·71-s − 0.225·79-s + 1/9·81-s + 2.11·89-s − 0.201·99-s + 0.796·101-s + 1.53·109-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.524482662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.524482662\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.617923166485020541039699362691, −9.188400291458291679037007918662, −8.721020722425421722943335958793, −8.090777049234184494344670427301, −7.56769400901303834898435326103, −7.31591507178608977707809656374, −6.56256714616671002927373963661, −6.10490391923692043188008510684, −5.37046128545327580986181494416, −4.97818173111463866326367880597, −4.34873025801747621573057796223, −3.54121389592464875097582431023, −2.95663321963890379833499266269, −2.13773733830389776980284506012, −0.983600206271662647349506315130,
0.983600206271662647349506315130, 2.13773733830389776980284506012, 2.95663321963890379833499266269, 3.54121389592464875097582431023, 4.34873025801747621573057796223, 4.97818173111463866326367880597, 5.37046128545327580986181494416, 6.10490391923692043188008510684, 6.56256714616671002927373963661, 7.31591507178608977707809656374, 7.56769400901303834898435326103, 8.090777049234184494344670427301, 8.721020722425421722943335958793, 9.188400291458291679037007918662, 9.617923166485020541039699362691
