| L(s) = 1 | − 3.36·3-s + 5-s − 1.19·7-s + 8.29·9-s + 2.16·11-s + 13-s − 3.36·15-s + 6·17-s + 2.16·19-s + 4·21-s + 7.70·23-s + 25-s − 17.7·27-s + 5.29·29-s + 4.55·31-s − 7.29·33-s − 1.19·35-s − 9.29·37-s − 3.36·39-s + 12.5·41-s − 10.0·43-s + 8.29·45-s − 1.19·47-s − 5.58·49-s − 20.1·51-s + 8.58·53-s + 2.16·55-s + ⋯ |
| L(s) = 1 | − 1.94·3-s + 0.447·5-s − 0.449·7-s + 2.76·9-s + 0.654·11-s + 0.277·13-s − 0.867·15-s + 1.45·17-s + 0.497·19-s + 0.872·21-s + 1.60·23-s + 0.200·25-s − 3.42·27-s + 0.982·29-s + 0.817·31-s − 1.26·33-s − 0.201·35-s − 1.52·37-s − 0.538·39-s + 1.96·41-s − 1.53·43-s + 1.23·45-s − 0.173·47-s − 0.797·49-s − 2.82·51-s + 1.17·53-s + 0.292·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.260282372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.260282372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.36T + 3T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 + 9.29T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 - 5.53T + 67T^{2} \) |
| 71 | \( 1 + 6.50T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 + 1.19T + 83T^{2} \) |
| 89 | \( 1 + 4.58T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.410659005567320113817760030031, −7.24871143531347297522734973386, −6.80163940859807825386697903257, −6.16695459476662888625136067879, −5.45139703480143022345338430303, −5.00046772720173259599794708494, −4.03315400870116633074188006048, −3.05258154946864322652578549970, −1.41366873843084005923733511954, −0.811652905642349258769064079498,
0.811652905642349258769064079498, 1.41366873843084005923733511954, 3.05258154946864322652578549970, 4.03315400870116633074188006048, 5.00046772720173259599794708494, 5.45139703480143022345338430303, 6.16695459476662888625136067879, 6.80163940859807825386697903257, 7.24871143531347297522734973386, 8.410659005567320113817760030031
