| L(s) = 1 | − 1.50·2-s − 0.544·3-s + 0.269·4-s + 3.40·5-s + 0.820·6-s − 0.714·7-s + 2.60·8-s − 2.70·9-s − 5.13·10-s + 11-s − 0.146·12-s + 1.07·14-s − 1.85·15-s − 4.46·16-s + 5.08·17-s + 4.07·18-s − 2.56·19-s + 0.916·20-s + 0.389·21-s − 1.50·22-s + 0.472·23-s − 1.42·24-s + 6.60·25-s + 3.10·27-s − 0.192·28-s + 7.13·29-s + 2.79·30-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.314·3-s + 0.134·4-s + 1.52·5-s + 0.335·6-s − 0.270·7-s + 0.921·8-s − 0.900·9-s − 1.62·10-s + 0.301·11-s − 0.0423·12-s + 0.287·14-s − 0.479·15-s − 1.11·16-s + 1.23·17-s + 0.959·18-s − 0.587·19-s + 0.205·20-s + 0.0850·21-s − 0.321·22-s + 0.0984·23-s − 0.290·24-s + 1.32·25-s + 0.598·27-s − 0.0363·28-s + 1.32·29-s + 0.510·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.000861163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.000861163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.50T + 2T^{2} \) |
| 3 | \( 1 + 0.544T + 3T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 7 | \( 1 + 0.714T + 7T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 0.472T + 23T^{2} \) |
| 29 | \( 1 - 7.13T + 29T^{2} \) |
| 31 | \( 1 + 7.93T + 31T^{2} \) |
| 37 | \( 1 - 1.68T + 37T^{2} \) |
| 41 | \( 1 - 1.29T + 41T^{2} \) |
| 43 | \( 1 - 9.09T + 43T^{2} \) |
| 47 | \( 1 - 7.45T + 47T^{2} \) |
| 53 | \( 1 - 7.78T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 7.36T + 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 + 6.44T + 73T^{2} \) |
| 79 | \( 1 + 5.05T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 8.30T + 89T^{2} \) |
| 97 | \( 1 - 9.02T + 97T^{2} \) |
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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
−9.089252660487759919712646046712, −8.860524811667130841480786614860, −7.82307396870555810911678267514, −6.92100332434447498057143576726, −5.95029859284415011610316428293, −5.54685772355505876952040298036, −4.43213675042604131861207035251, −2.99814128810223559209470092209, −1.93594480763767341332971238112, −0.832239380529918244876856604501,
0.832239380529918244876856604501, 1.93594480763767341332971238112, 2.99814128810223559209470092209, 4.43213675042604131861207035251, 5.54685772355505876952040298036, 5.95029859284415011610316428293, 6.92100332434447498057143576726, 7.82307396870555810911678267514, 8.860524811667130841480786614860, 9.089252660487759919712646046712
