| L(s) = 1 | − 4.46·2-s − 9.80·3-s + 11.9·4-s + 43.8·6-s − 17.6·8-s + 69.2·9-s − 56.5·11-s − 117.·12-s − 40.9·13-s − 16.6·16-s − 2.18·17-s − 309.·18-s + 16.4·19-s + 252.·22-s + 155.·23-s + 173.·24-s + 183.·26-s − 414.·27-s − 6.26·29-s + 168.·31-s + 215.·32-s + 554.·33-s + 9.77·34-s + 827.·36-s + 37.1·37-s − 73.5·38-s + 401.·39-s + ⋯ |
| L(s) = 1 | − 1.57·2-s − 1.88·3-s + 1.49·4-s + 2.98·6-s − 0.781·8-s + 2.56·9-s − 1.54·11-s − 2.82·12-s − 0.873·13-s − 0.260·16-s − 0.0312·17-s − 4.04·18-s + 0.198·19-s + 2.44·22-s + 1.40·23-s + 1.47·24-s + 1.38·26-s − 2.95·27-s − 0.0400·29-s + 0.977·31-s + 1.19·32-s + 2.92·33-s + 0.0493·34-s + 3.83·36-s + 0.165·37-s − 0.314·38-s + 1.64·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1793303900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1793303900\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.46T + 8T^{2} \) |
| 3 | \( 1 + 9.80T + 27T^{2} \) |
| 11 | \( 1 + 56.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.18T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6.26T + 2.43e4T^{2} \) |
| 31 | \( 1 - 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 14.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 151.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 934.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 300.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 752.T + 9.12e5T^{2} \) |
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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
−9.692820065015155382887766360029, −8.567343673942010613765602179995, −7.49402314174017110490480144945, −7.15926201574515694415100190981, −6.18469143469699467593127202651, −5.20859275122217297220934997058, −4.63836806820666409382779166302, −2.64903046175087497124700479832, −1.32226627112145691779050048774, −0.33304368055917984196650067721,
0.33304368055917984196650067721, 1.32226627112145691779050048774, 2.64903046175087497124700479832, 4.63836806820666409382779166302, 5.20859275122217297220934997058, 6.18469143469699467593127202651, 7.15926201574515694415100190981, 7.49402314174017110490480144945, 8.567343673942010613765602179995, 9.692820065015155382887766360029
