| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s − 11-s + 15-s − 6·17-s − 4·19-s − 3·21-s − 4·23-s + 25-s + 27-s − 7·31-s − 33-s − 3·35-s − 10·37-s + 7·43-s + 45-s + 2·49-s − 6·51-s − 2·53-s − 55-s − 4·57-s − 15·61-s − 3·63-s − 13·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.25·31-s − 0.174·33-s − 0.507·35-s − 1.64·37-s + 1.06·43-s + 0.149·45-s + 2/7·49-s − 0.840·51-s − 0.274·53-s − 0.134·55-s − 0.529·57-s − 1.92·61-s − 0.377·63-s − 1.58·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
| 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
−13.97402477965719, −13.63586866861416, −13.17122321843646, −12.75655720585252, −12.41136185582203, −11.80670768227540, −11.00301052082372, −10.63671315804252, −10.24277560946303, −9.654204234976141, −9.103682453802379, −8.880129307148902, −8.347991962741308, −7.600157214723422, −7.098714117978900, −6.672639340147667, −6.039565674286106, −5.760027962658112, −4.877418767242925, −4.280056273393892, −3.852257997603028, −3.078920346856852, −2.680176653516808, −1.960144711982613, −1.520251158298851, 0, 0,
1.520251158298851, 1.960144711982613, 2.680176653516808, 3.078920346856852, 3.852257997603028, 4.280056273393892, 4.877418767242925, 5.760027962658112, 6.039565674286106, 6.672639340147667, 7.098714117978900, 7.600157214723422, 8.347991962741308, 8.880129307148902, 9.103682453802379, 9.654204234976141, 10.24277560946303, 10.63671315804252, 11.00301052082372, 11.80670768227540, 12.41136185582203, 12.75655720585252, 13.17122321843646, 13.63586866861416, 13.97402477965719
