| L(s) = 1 | + 6.32·3-s + 17.8·5-s − 22.6·7-s + 13.0·9-s + 44.2·11-s + 17.8·13-s + 113.·15-s + 70·17-s + 82.2·19-s − 143.·21-s − 158.·23-s + 195.·25-s − 88.5·27-s − 125.·29-s + 280·33-s − 404.·35-s + 375.·37-s + 113.·39-s + 182·41-s − 132.·43-s + 232.·45-s − 316.·47-s + 169.·49-s + 442.·51-s − 125.·53-s + 791.·55-s + 520·57-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 1.60·5-s − 1.22·7-s + 0.481·9-s + 1.21·11-s + 0.381·13-s + 1.94·15-s + 0.998·17-s + 0.992·19-s − 1.48·21-s − 1.43·23-s + 1.56·25-s − 0.631·27-s − 0.801·29-s + 1.47·33-s − 1.95·35-s + 1.66·37-s + 0.464·39-s + 0.693·41-s − 0.471·43-s + 0.770·45-s − 0.983·47-s + 0.492·49-s + 1.21·51-s − 0.324·53-s + 1.94·55-s + 1.20·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.323084619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.323084619\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 6.32T + 27T^{2} \) |
| 5 | \( 1 - 17.8T + 125T^{2} \) |
| 7 | \( 1 + 22.6T + 343T^{2} \) |
| 11 | \( 1 - 44.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 - 375.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 182T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 316.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 125.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 82.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 221.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 113.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 910T + 3.89e5T^{2} \) |
| 79 | \( 1 + 678.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 714.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 546T + 7.04e5T^{2} \) |
| 97 | \( 1 + 490T + 9.12e5T^{2} \) |
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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
−11.67337385182585094769958386276, −10.02119051332257663443745078699, −9.597678755215540686089111384142, −9.033809973051537377322725435369, −7.74066911150519706963832341416, −6.39289495823577702074046085854, −5.75538755066282345655153834152, −3.76722166643246114933524129058, −2.80810883829530013745395077812, −1.51754484392689165697370344442,
1.51754484392689165697370344442, 2.80810883829530013745395077812, 3.76722166643246114933524129058, 5.75538755066282345655153834152, 6.39289495823577702074046085854, 7.74066911150519706963832341416, 9.033809973051537377322725435369, 9.597678755215540686089111384142, 10.02119051332257663443745078699, 11.67337385182585094769958386276
