| L(s) = 1 | − 3-s − 4·7-s + 9-s − 13-s + 17-s + 2·19-s + 4·21-s − 5·25-s − 27-s + 2·29-s + 8·37-s + 39-s + 4·41-s − 4·43-s − 2·47-s + 9·49-s − 51-s − 6·53-s − 2·57-s − 6·59-s + 6·61-s − 4·63-s + 2·67-s + 5·75-s − 8·79-s + 81-s + 2·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.242·17-s + 0.458·19-s + 0.872·21-s − 25-s − 0.192·27-s + 0.371·29-s + 1.31·37-s + 0.160·39-s + 0.624·41-s − 0.609·43-s − 0.291·47-s + 9/7·49-s − 0.140·51-s − 0.824·53-s − 0.264·57-s − 0.781·59-s + 0.768·61-s − 0.503·63-s + 0.244·67-s + 0.577·75-s − 0.900·79-s + 1/9·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9716248013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9716248013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.71457285630850, −12.03324147632738, −11.90697470390616, −11.24040163099333, −10.87667992089925, −10.17873209974291, −9.913385363497261, −9.519383271317379, −9.209324206995122, −8.462401651979755, −7.935537900478138, −7.442159097634523, −7.012114294218143, −6.399306093404742, −6.164373820456890, −5.663259832696918, −5.158769656424115, −4.487401388969493, −4.059765144465508, −3.401066205110604, −3.017966542909509, −2.422576856374139, −1.704443603996042, −0.9225664745386388, −0.3234111090829367,
0.3234111090829367, 0.9225664745386388, 1.704443603996042, 2.422576856374139, 3.017966542909509, 3.401066205110604, 4.059765144465508, 4.487401388969493, 5.158769656424115, 5.663259832696918, 6.164373820456890, 6.399306093404742, 7.012114294218143, 7.442159097634523, 7.935537900478138, 8.462401651979755, 9.209324206995122, 9.519383271317379, 9.913385363497261, 10.17873209974291, 10.87667992089925, 11.24040163099333, 11.90697470390616, 12.03324147632738, 12.71457285630850
