| L(s) = 1 | + 2.29e4·2-s + 4.78e6·3-s − 1.13e7·4-s − 1.77e10·5-s + 1.09e11·6-s − 6.78e11·7-s − 1.25e13·8-s + 2.28e13·9-s − 4.07e14·10-s − 2.01e15·11-s − 5.40e13·12-s − 5.13e15·13-s − 1.55e16·14-s − 8.50e16·15-s − 2.82e17·16-s + 4.65e17·17-s + 5.24e17·18-s + 5.09e17·19-s + 2.01e17·20-s − 3.24e18·21-s − 4.62e19·22-s + 3.60e19·23-s − 6.01e19·24-s + 1.29e20·25-s − 1.17e20·26-s + 1.09e20·27-s + 7.66e18·28-s + ⋯ |
| L(s) = 1 | + 0.989·2-s + 0.577·3-s − 0.0210·4-s − 1.30·5-s + 0.571·6-s − 0.377·7-s − 1.01·8-s + 0.333·9-s − 1.28·10-s − 1.60·11-s − 0.0121·12-s − 0.361·13-s − 0.373·14-s − 0.752·15-s − 0.978·16-s + 0.670·17-s + 0.329·18-s + 0.146·19-s + 0.0274·20-s − 0.218·21-s − 1.58·22-s + 0.648·23-s − 0.583·24-s + 0.697·25-s − 0.357·26-s + 0.192·27-s + 0.00796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.574968396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.574968396\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4.78e6T \) |
| 7 | \( 1 + 6.78e11T \) |
| good | 2 | \( 1 - 2.29e4T + 5.36e8T^{2} \) |
| 5 | \( 1 + 1.77e10T + 1.86e20T^{2} \) |
| 11 | \( 1 + 2.01e15T + 1.58e30T^{2} \) |
| 13 | \( 1 + 5.13e15T + 2.01e32T^{2} \) |
| 17 | \( 1 - 4.65e17T + 4.81e35T^{2} \) |
| 19 | \( 1 - 5.09e17T + 1.21e37T^{2} \) |
| 23 | \( 1 - 3.60e19T + 3.09e39T^{2} \) |
| 29 | \( 1 + 9.09e20T + 2.56e42T^{2} \) |
| 31 | \( 1 + 3.68e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 1.06e23T + 3.00e45T^{2} \) |
| 41 | \( 1 - 2.67e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 5.83e23T + 2.34e47T^{2} \) |
| 47 | \( 1 - 1.05e24T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.57e25T + 1.00e50T^{2} \) |
| 59 | \( 1 - 7.97e25T + 2.26e51T^{2} \) |
| 61 | \( 1 - 5.17e25T + 5.95e51T^{2} \) |
| 67 | \( 1 + 6.66e25T + 9.04e52T^{2} \) |
| 71 | \( 1 + 8.22e26T + 4.85e53T^{2} \) |
| 73 | \( 1 - 9.66e26T + 1.08e54T^{2} \) |
| 79 | \( 1 - 2.57e27T + 1.07e55T^{2} \) |
| 83 | \( 1 + 9.93e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + 2.67e28T + 3.40e56T^{2} \) |
| 97 | \( 1 + 1.58e28T + 4.13e57T^{2} \) |
| 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
−12.56531704176382223546845148094, −11.24629597742101933169964679228, −9.656994507741302645229387968172, −8.231587739341893235376119389916, −7.33226668158338466753893215742, −5.55200450448101713679223229301, −4.45361617086168375833717150547, −3.43400247332639909406621769973, −2.64269765425654118718281932667, −0.47216541919832652356399031894,
0.47216541919832652356399031894, 2.64269765425654118718281932667, 3.43400247332639909406621769973, 4.45361617086168375833717150547, 5.55200450448101713679223229301, 7.33226668158338466753893215742, 8.231587739341893235376119389916, 9.656994507741302645229387968172, 11.24629597742101933169964679228, 12.56531704176382223546845148094
