| L(s) = 1 | − 0.967·2-s − 3-s − 1.06·4-s + 5-s + 0.967·6-s + 0.0385·7-s + 2.96·8-s + 9-s − 0.967·10-s + 1.36·11-s + 1.06·12-s − 5.05·13-s − 0.0373·14-s − 15-s − 0.742·16-s − 8.03·17-s − 0.967·18-s − 6.83·19-s − 1.06·20-s − 0.0385·21-s − 1.32·22-s − 6.37·23-s − 2.96·24-s + 25-s + 4.89·26-s − 27-s − 0.0410·28-s + ⋯ |
| L(s) = 1 | − 0.684·2-s − 0.577·3-s − 0.531·4-s + 0.447·5-s + 0.395·6-s + 0.0145·7-s + 1.04·8-s + 0.333·9-s − 0.306·10-s + 0.412·11-s + 0.306·12-s − 1.40·13-s − 0.00997·14-s − 0.258·15-s − 0.185·16-s − 1.94·17-s − 0.228·18-s − 1.56·19-s − 0.237·20-s − 0.00841·21-s − 0.282·22-s − 1.32·23-s − 0.605·24-s + 0.200·25-s + 0.960·26-s − 0.192·27-s − 0.00774·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3648792201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3648792201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.967T + 2T^{2} \) |
| 7 | \( 1 - 0.0385T + 7T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 + 8.03T + 17T^{2} \) |
| 19 | \( 1 + 6.83T + 19T^{2} \) |
| 23 | \( 1 + 6.37T + 23T^{2} \) |
| 29 | \( 1 + 1.62T + 29T^{2} \) |
| 31 | \( 1 - 8.44T + 31T^{2} \) |
| 37 | \( 1 + 1.79T + 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 + 8.81T + 43T^{2} \) |
| 47 | \( 1 + 0.604T + 47T^{2} \) |
| 53 | \( 1 - 5.09T + 53T^{2} \) |
| 59 | \( 1 + 7.44T + 59T^{2} \) |
| 61 | \( 1 + 0.111T + 61T^{2} \) |
| 67 | \( 1 + 1.06T + 67T^{2} \) |
| 71 | \( 1 + 0.673T + 71T^{2} \) |
| 73 | \( 1 - 7.61T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.68T + 83T^{2} \) |
| 89 | \( 1 + 0.405T + 89T^{2} \) |
| 97 | \( 1 - 2.63T + 97T^{2} \) |
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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
−8.227555010128336247458096061834, −7.43015115451572338905034447440, −6.54095504126574018275000331789, −6.23329538348187614907142493705, −4.97093192811466277975545602732, −4.61871977569963717523652672006, −3.92681899178356775394858696472, −2.37409518918166640068296458676, −1.79725191473328692970476655997, −0.35362292723897099619370145760,
0.35362292723897099619370145760, 1.79725191473328692970476655997, 2.37409518918166640068296458676, 3.92681899178356775394858696472, 4.61871977569963717523652672006, 4.97093192811466277975545602732, 6.23329538348187614907142493705, 6.54095504126574018275000331789, 7.43015115451572338905034447440, 8.227555010128336247458096061834
