| L(s) = 1 | + 2.56·2-s − 1.56·3-s + 4.56·4-s − 1.56·5-s − 4·6-s + 5.12·7-s + 6.56·8-s − 0.561·9-s − 4·10-s + 0.438·11-s − 7.12·12-s + 13-s + 13.1·14-s + 2.43·15-s + 7.68·16-s − 6·17-s − 1.43·18-s + 0.438·19-s − 7.12·20-s − 8·21-s + 1.12·22-s − 23-s − 10.2·24-s − 2.56·25-s + 2.56·26-s + 5.56·27-s + 23.3·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 0.901·3-s + 2.28·4-s − 0.698·5-s − 1.63·6-s + 1.93·7-s + 2.31·8-s − 0.187·9-s − 1.26·10-s + 0.132·11-s − 2.05·12-s + 0.277·13-s + 3.50·14-s + 0.629·15-s + 1.92·16-s − 1.45·17-s − 0.339·18-s + 0.100·19-s − 1.59·20-s − 1.74·21-s + 0.239·22-s − 0.208·23-s − 2.09·24-s − 0.512·25-s + 0.502·26-s + 1.07·27-s + 4.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.739829342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.739829342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 0.438T + 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 0.438T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 0.876T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 2.68T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 - 2.68T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
| 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
−11.67453237373255852097282427518, −11.25974335317578081804164930303, −10.82370731651051683670828708182, −8.575332756975851289581821031551, −7.53151364260431552591408628346, −6.46013664807544657445831993770, −5.36921483953886945628504627033, −4.75223925291913593156048982289, −3.83790865728607445507592039672, −2.02221965148494179126227516215,
2.02221965148494179126227516215, 3.83790865728607445507592039672, 4.75223925291913593156048982289, 5.36921483953886945628504627033, 6.46013664807544657445831993770, 7.53151364260431552591408628346, 8.575332756975851289581821031551, 10.82370731651051683670828708182, 11.25974335317578081804164930303, 11.67453237373255852097282427518
