| L(s) = 1 | − 3.59·5-s − 5.06·7-s + 2.55·11-s − 4.93·13-s − 2.68·17-s + 4.72·19-s + 1.49·23-s + 7.93·25-s − 2.43·29-s + 3.19·31-s + 18.2·35-s + 2.90·37-s + 41-s − 7.62·43-s − 5.15·47-s + 18.6·49-s + 11.1·53-s − 9.20·55-s − 1.49·59-s + 1.06·61-s + 17.7·65-s + 10.5·67-s + 12.6·71-s + 8.28·73-s − 12.9·77-s − 4.28·79-s − 10.5·83-s + ⋯ |
| L(s) = 1 | − 1.60·5-s − 1.91·7-s + 0.771·11-s − 1.36·13-s − 0.651·17-s + 1.08·19-s + 0.311·23-s + 1.58·25-s − 0.451·29-s + 0.573·31-s + 3.07·35-s + 0.478·37-s + 0.156·41-s − 1.16·43-s − 0.751·47-s + 2.66·49-s + 1.52·53-s − 1.24·55-s − 0.194·59-s + 0.136·61-s + 2.20·65-s + 1.29·67-s + 1.49·71-s + 0.969·73-s − 1.47·77-s − 0.482·79-s − 1.15·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6088191262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6088191262\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 + 3.59T + 5T^{2} \) |
| 7 | \( 1 + 5.06T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 + 2.43T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 1.49T + 59T^{2} \) |
| 61 | \( 1 - 1.06T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 8.28T + 73T^{2} \) |
| 79 | \( 1 + 4.28T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
| 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
−9.578027059829328317975120419353, −8.766011523834010565818362508625, −7.76582901645540745796160781321, −6.98010860881313725832975491723, −6.61026385980946124203808388269, −5.27954329773529416575941485120, −4.18870140132638145145953870990, −3.50503510240618136130009225147, −2.70839483449602991472872426620, −0.52754037486423125027462925060,
0.52754037486423125027462925060, 2.70839483449602991472872426620, 3.50503510240618136130009225147, 4.18870140132638145145953870990, 5.27954329773529416575941485120, 6.61026385980946124203808388269, 6.98010860881313725832975491723, 7.76582901645540745796160781321, 8.766011523834010565818362508625, 9.578027059829328317975120419353
