| L(s) = 1 | − 1.25·2-s + 2.51·3-s − 0.430·4-s + 5-s − 3.14·6-s + 0.281·7-s + 3.04·8-s + 3.31·9-s − 1.25·10-s − 0.160·11-s − 1.08·12-s + 1.61·13-s − 0.353·14-s + 2.51·15-s − 2.95·16-s − 2.88·17-s − 4.15·18-s + 4.15·19-s − 0.430·20-s + 0.708·21-s + 0.201·22-s − 2.52·23-s + 7.65·24-s + 25-s − 2.02·26-s + 0.800·27-s − 0.121·28-s + ⋯ |
| L(s) = 1 | − 0.885·2-s + 1.45·3-s − 0.215·4-s + 0.447·5-s − 1.28·6-s + 0.106·7-s + 1.07·8-s + 1.10·9-s − 0.396·10-s − 0.0483·11-s − 0.312·12-s + 0.448·13-s − 0.0943·14-s + 0.649·15-s − 0.738·16-s − 0.700·17-s − 0.979·18-s + 0.952·19-s − 0.0963·20-s + 0.154·21-s + 0.0428·22-s − 0.525·23-s + 1.56·24-s + 0.200·25-s − 0.397·26-s + 0.154·27-s − 0.0229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.152805802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152805802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 3 | \( 1 - 2.51T + 3T^{2} \) |
| 7 | \( 1 - 0.281T + 7T^{2} \) |
| 11 | \( 1 + 0.160T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 - 2.70T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 5.88T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 6.86T + 53T^{2} \) |
| 59 | \( 1 + 5.42T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 0.701T + 73T^{2} \) |
| 79 | \( 1 + 1.24T + 79T^{2} \) |
| 83 | \( 1 - 0.157T + 83T^{2} \) |
| 89 | \( 1 + 7.21T + 89T^{2} \) |
| 97 | \( 1 - 6.79T + 97T^{2} \) |
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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
−13.00354758848694874460729451288, −11.39998908261262667868338243042, −10.05273858215198829297021985609, −9.548844496640522815746456382345, −8.488899932100994967980573507477, −8.085135066883956402718523758228, −6.74824623233288842945439857297, −4.86312334539955329431812453176, −3.37570481018297562284149139298, −1.78334349113768273573355120241,
1.78334349113768273573355120241, 3.37570481018297562284149139298, 4.86312334539955329431812453176, 6.74824623233288842945439857297, 8.085135066883956402718523758228, 8.488899932100994967980573507477, 9.548844496640522815746456382345, 10.05273858215198829297021985609, 11.39998908261262667868338243042, 13.00354758848694874460729451288
