| L(s) = 1 | − 2.19·3-s − 7-s + 1.83·9-s + 1.37·11-s − 2.74·13-s − 6.94·17-s − 1.29·19-s + 2.19·21-s − 8.31·23-s + 2.56·27-s − 8.40·29-s + 9.49·31-s − 3.02·33-s − 1.73·37-s + 6.02·39-s − 5.30·41-s + 7.83·43-s + 3.48·47-s + 49-s + 15.2·51-s − 6.13·53-s + 2.83·57-s − 6.26·59-s + 6.59·61-s − 1.83·63-s + 1.66·67-s + 18.2·69-s + ⋯ |
| L(s) = 1 | − 1.26·3-s − 0.377·7-s + 0.610·9-s + 0.414·11-s − 0.760·13-s − 1.68·17-s − 0.296·19-s + 0.479·21-s − 1.73·23-s + 0.494·27-s − 1.56·29-s + 1.70·31-s − 0.525·33-s − 0.285·37-s + 0.964·39-s − 0.829·41-s + 1.19·43-s + 0.508·47-s + 0.142·49-s + 2.13·51-s − 0.842·53-s + 0.375·57-s − 0.815·59-s + 0.844·61-s − 0.230·63-s + 0.203·67-s + 2.20·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4192715489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4192715489\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 2.19T + 3T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2.74T + 13T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 19 | \( 1 + 1.29T + 19T^{2} \) |
| 23 | \( 1 + 8.31T + 23T^{2} \) |
| 29 | \( 1 + 8.40T + 29T^{2} \) |
| 31 | \( 1 - 9.49T + 31T^{2} \) |
| 37 | \( 1 + 1.73T + 37T^{2} \) |
| 41 | \( 1 + 5.30T + 41T^{2} \) |
| 43 | \( 1 - 7.83T + 43T^{2} \) |
| 47 | \( 1 - 3.48T + 47T^{2} \) |
| 53 | \( 1 + 6.13T + 53T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 - 6.59T + 61T^{2} \) |
| 67 | \( 1 - 1.66T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 2.13T + 73T^{2} \) |
| 79 | \( 1 - 5.45T + 79T^{2} \) |
| 83 | \( 1 - 2.54T + 83T^{2} \) |
| 89 | \( 1 - 1.43T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
| 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
−8.069677909840206658381026536916, −7.23328928652653043971758275665, −6.47029939184953342438663482311, −6.14838422683907144631913679784, −5.33508687765564903804383900348, −4.50574186261911905798069392165, −3.99353098640246013194934149256, −2.69667887694498289936877125006, −1.80034347335116663175778603044, −0.35409550608324087215259992926,
0.35409550608324087215259992926, 1.80034347335116663175778603044, 2.69667887694498289936877125006, 3.99353098640246013194934149256, 4.50574186261911905798069392165, 5.33508687765564903804383900348, 6.14838422683907144631913679784, 6.47029939184953342438663482311, 7.23328928652653043971758275665, 8.069677909840206658381026536916
