| L(s) = 1 | + 3·3-s + 12.9·5-s − 4.19·7-s + 9·9-s − 64.0·11-s + 82.4·13-s + 38.7·15-s + 71.6·17-s − 23.7·19-s − 12.5·21-s + 23·23-s + 42.1·25-s + 27·27-s + 264.·29-s + 239.·31-s − 192.·33-s − 54.1·35-s + 229.·37-s + 247.·39-s − 230.·41-s − 46.3·43-s + 116.·45-s − 119.·47-s − 325.·49-s + 214.·51-s + 122.·53-s − 827.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.15·5-s − 0.226·7-s + 0.333·9-s − 1.75·11-s + 1.75·13-s + 0.667·15-s + 1.02·17-s − 0.287·19-s − 0.130·21-s + 0.208·23-s + 0.336·25-s + 0.192·27-s + 1.69·29-s + 1.38·31-s − 1.01·33-s − 0.261·35-s + 1.01·37-s + 1.01·39-s − 0.878·41-s − 0.164·43-s + 0.385·45-s − 0.371·47-s − 0.948·49-s + 0.590·51-s + 0.316·53-s − 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.081232926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.081232926\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 12.9T + 125T^{2} \) |
| 7 | \( 1 + 4.19T + 343T^{2} \) |
| 11 | \( 1 + 64.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 264.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 239.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 229.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 230.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 46.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 119.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 647.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 835.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 24.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 411.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 961.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 246.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 822.T + 9.12e5T^{2} \) |
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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
−10.20907584596019042397725309878, −9.684918245682083683215856515496, −8.434983692544542826579239594833, −8.034326513951258224619696268800, −6.58052562962340734202120744015, −5.84084144201368171959989024948, −4.83857560148198442364944876953, −3.32372811440652477095727036217, −2.45061885470856272137945089875, −1.11552220010468724056344332851,
1.11552220010468724056344332851, 2.45061885470856272137945089875, 3.32372811440652477095727036217, 4.83857560148198442364944876953, 5.84084144201368171959989024948, 6.58052562962340734202120744015, 8.034326513951258224619696268800, 8.434983692544542826579239594833, 9.684918245682083683215856515496, 10.20907584596019042397725309878
