L(s) = 1 | − 2.49·5-s − 0.917·7-s + 3.69·11-s + 5.81·13-s + 0.867·17-s − 6.52·19-s + 4·23-s + 1.23·25-s + 7.72·29-s + 2.14·31-s + 2.29·35-s − 2.47·37-s + 9.58·41-s + 9.58·43-s − 1.65·47-s − 6.15·49-s − 3.39·53-s − 9.22·55-s − 12.7·59-s + 0.0231·61-s − 14.5·65-s − 5.32·67-s + 11.8·71-s − 15.2·73-s − 3.39·77-s − 8.40·79-s + 1.96·83-s + ⋯ |
L(s) = 1 | − 1.11·5-s − 0.346·7-s + 1.11·11-s + 1.61·13-s + 0.210·17-s − 1.49·19-s + 0.834·23-s + 0.246·25-s + 1.43·29-s + 0.385·31-s + 0.387·35-s − 0.406·37-s + 1.49·41-s + 1.46·43-s − 0.241·47-s − 0.879·49-s − 0.466·53-s − 1.24·55-s − 1.65·59-s + 0.00296·61-s − 1.79·65-s − 0.650·67-s + 1.40·71-s − 1.78·73-s − 0.386·77-s − 0.946·79-s + 0.215·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.741110718\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.741110718\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 + 0.917T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 - 5.81T + 13T^{2} \) |
| 17 | \( 1 - 0.867T + 17T^{2} \) |
| 19 | \( 1 + 6.52T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 7.72T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 - 9.58T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 0.0231T + 61T^{2} \) |
| 67 | \( 1 + 5.32T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 8.40T + 79T^{2} \) |
| 83 | \( 1 - 1.96T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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
−7.77865407501242598779243923214, −6.99527839554370994107567780625, −6.27196282247380667183044383415, −6.01145197863509549097431991359, −4.60899758577858422020376151381, −4.21823403614216778001820162022, −3.54095738023273971262047762140, −2.85167040866591371522592800218, −1.53615073348293259275981400294, −0.67738122922844691886874587425,
0.67738122922844691886874587425, 1.53615073348293259275981400294, 2.85167040866591371522592800218, 3.54095738023273971262047762140, 4.21823403614216778001820162022, 4.60899758577858422020376151381, 6.01145197863509549097431991359, 6.27196282247380667183044383415, 6.99527839554370994107567780625, 7.77865407501242598779243923214