| L(s) = 1 | + 0.604·3-s + 0.604·5-s + 7-s − 2.63·9-s + 1.77·11-s + 3.40·13-s + 0.365·15-s + 5.22·17-s + 4·19-s + 0.604·21-s − 23-s − 4.63·25-s − 3.40·27-s − 5.17·29-s + 9.87·31-s + 1.07·33-s + 0.604·35-s − 5.82·37-s + 2.05·39-s − 2·41-s − 1.40·43-s − 1.59·45-s + 3.81·47-s + 49-s + 3.15·51-s − 2.65·53-s + 1.07·55-s + ⋯ |
| L(s) = 1 | + 0.349·3-s + 0.270·5-s + 0.377·7-s − 0.878·9-s + 0.534·11-s + 0.944·13-s + 0.0943·15-s + 1.26·17-s + 0.917·19-s + 0.131·21-s − 0.208·23-s − 0.926·25-s − 0.655·27-s − 0.961·29-s + 1.77·31-s + 0.186·33-s + 0.102·35-s − 0.957·37-s + 0.329·39-s − 0.312·41-s − 0.214·43-s − 0.237·45-s + 0.556·47-s + 0.142·49-s + 0.441·51-s − 0.364·53-s + 0.144·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.389786454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.389786454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.604T + 3T^{2} \) |
| 5 | \( 1 - 0.604T + 5T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 - 3.40T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 5.17T + 29T^{2} \) |
| 31 | \( 1 - 9.87T + 31T^{2} \) |
| 37 | \( 1 + 5.82T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 1.40T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 + 2.65T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 6.66T + 61T^{2} \) |
| 67 | \( 1 + 5.04T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6.02T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 8.75T + 83T^{2} \) |
| 89 | \( 1 + 0.466T + 89T^{2} \) |
| 97 | \( 1 + 3.02T + 97T^{2} \) |
| 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
−8.810087868067088481056520521360, −8.167823951230071406423675472575, −7.56919363737672025763011133136, −6.48010243481892832941722518224, −5.75138174330283904939394190143, −5.14251503131717024262528425628, −3.83795319831218518426584981026, −3.27900537315997217630470410704, −2.11308872499037030189807825237, −1.01504258379084369243979172181,
1.01504258379084369243979172181, 2.11308872499037030189807825237, 3.27900537315997217630470410704, 3.83795319831218518426584981026, 5.14251503131717024262528425628, 5.75138174330283904939394190143, 6.48010243481892832941722518224, 7.56919363737672025763011133136, 8.167823951230071406423675472575, 8.810087868067088481056520521360
