L(s) = 1 | − 0.414·2-s − 1.41·3-s − 1.82·4-s + 0.585·6-s + 0.828·7-s + 1.58·8-s − 0.999·9-s + 0.585·11-s + 2.58·12-s + 13-s − 0.343·14-s + 3·16-s + 4.82·17-s + 0.414·18-s + 3.41·19-s − 1.17·21-s − 0.242·22-s + 1.41·23-s − 2.24·24-s − 0.414·26-s + 5.65·27-s − 1.51·28-s + 5.65·29-s + 10.2·31-s − 4.41·32-s − 0.828·33-s − 1.99·34-s + ⋯ |
L(s) = 1 | − 0.292·2-s − 0.816·3-s − 0.914·4-s + 0.239·6-s + 0.313·7-s + 0.560·8-s − 0.333·9-s + 0.176·11-s + 0.746·12-s + 0.277·13-s − 0.0917·14-s + 0.750·16-s + 1.17·17-s + 0.0976·18-s + 0.783·19-s − 0.255·21-s − 0.0517·22-s + 0.294·23-s − 0.457·24-s − 0.0812·26-s + 1.08·27-s − 0.286·28-s + 1.05·29-s + 1.83·31-s − 0.780·32-s − 0.144·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7201271166\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7201271166\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 - 0.585T + 11T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 3.07T + 43T^{2} \) |
| 47 | \( 1 + 0.828T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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
−11.77870310231878914070473377692, −10.51439145183168389682277154383, −9.921831006714229476653878663441, −8.707110158805151599903507373851, −8.045988069249354466622619321418, −6.69923811262987715970847853291, −5.48279351812368746279929448582, −4.82254829971895866155224576747, −3.35384825351936478667464228520, −0.985633272916172848437581855252,
0.985633272916172848437581855252, 3.35384825351936478667464228520, 4.82254829971895866155224576747, 5.48279351812368746279929448582, 6.69923811262987715970847853291, 8.045988069249354466622619321418, 8.707110158805151599903507373851, 9.921831006714229476653878663441, 10.51439145183168389682277154383, 11.77870310231878914070473377692