| L(s) = 1 | + 3-s − 0.766·5-s − 1.64·7-s + 9-s − 2.03·11-s − 4.75·13-s − 0.766·15-s + 3.00·17-s + 3.50·19-s − 1.64·21-s − 0.192·23-s − 4.41·25-s + 27-s + 7.19·29-s + 10.3·31-s − 2.03·33-s + 1.26·35-s − 7.01·37-s − 4.75·39-s − 8.36·41-s + 7.87·43-s − 0.766·45-s + 1.15·47-s − 4.28·49-s + 3.00·51-s − 1.52·53-s + 1.56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.342·5-s − 0.622·7-s + 0.333·9-s − 0.614·11-s − 1.31·13-s − 0.198·15-s + 0.728·17-s + 0.805·19-s − 0.359·21-s − 0.0401·23-s − 0.882·25-s + 0.192·27-s + 1.33·29-s + 1.86·31-s − 0.354·33-s + 0.213·35-s − 1.15·37-s − 0.761·39-s − 1.30·41-s + 1.20·43-s − 0.114·45-s + 0.168·47-s − 0.612·49-s + 0.420·51-s − 0.209·53-s + 0.210·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.727207621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727207621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 + 0.766T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 + 4.75T + 13T^{2} \) |
| 17 | \( 1 - 3.00T + 17T^{2} \) |
| 19 | \( 1 - 3.50T + 19T^{2} \) |
| 23 | \( 1 + 0.192T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.01T + 37T^{2} \) |
| 41 | \( 1 + 8.36T + 41T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 - 1.15T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 - 3.96T + 67T^{2} \) |
| 71 | \( 1 + 9.75T + 71T^{2} \) |
| 73 | \( 1 + 2.35T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 5.97T + 83T^{2} \) |
| 89 | \( 1 - 7.45T + 89T^{2} \) |
| 97 | \( 1 - 4.21T + 97T^{2} \) |
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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
−7.928069876337917295963398962897, −7.55156402996424607230299491839, −6.79020312926472591904154842906, −6.00208992418169548473397707412, −5.03289182952433189779624008840, −4.52293900292205725412575586750, −3.35262807188802273255963084334, −2.96742006089613046231643705363, −2.02244884351820177867138679818, −0.64985548401346656272345741140,
0.64985548401346656272345741140, 2.02244884351820177867138679818, 2.96742006089613046231643705363, 3.35262807188802273255963084334, 4.52293900292205725412575586750, 5.03289182952433189779624008840, 6.00208992418169548473397707412, 6.79020312926472591904154842906, 7.55156402996424607230299491839, 7.928069876337917295963398962897
