| L(s) = 1 | − 3·3-s + 6.95·5-s + 9·9-s + 43.9·11-s + 83.5·13-s − 20.8·15-s − 10.4·17-s − 4.27·19-s + 160.·23-s − 76.5·25-s − 27·27-s + 9.93·29-s + 133.·31-s − 131.·33-s − 357.·37-s − 250.·39-s + 127.·41-s + 343.·43-s + 62.6·45-s − 77.4·47-s + 31.3·51-s − 460.·53-s + 305.·55-s + 12.8·57-s − 272.·59-s + 51.3·61-s + 581.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.622·5-s + 0.333·9-s + 1.20·11-s + 1.78·13-s − 0.359·15-s − 0.148·17-s − 0.0516·19-s + 1.45·23-s − 0.612·25-s − 0.192·27-s + 0.0635·29-s + 0.774·31-s − 0.694·33-s − 1.58·37-s − 1.02·39-s + 0.484·41-s + 1.21·43-s + 0.207·45-s − 0.240·47-s + 0.0859·51-s − 1.19·53-s + 0.749·55-s + 0.0298·57-s − 0.601·59-s + 0.107·61-s + 1.10·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.481246664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481246664\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6.95T + 125T^{2} \) |
| 11 | \( 1 - 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 83.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4.27T + 6.85e3T^{2} \) |
| 23 | \( 1 - 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 9.93T + 2.43e4T^{2} \) |
| 31 | \( 1 - 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 77.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 460.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 272.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 51.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 327.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 571.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 206.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 923.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 97.6T + 9.12e5T^{2} \) |
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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
−9.197534866637912540350401812402, −8.901296996395076179144369887469, −7.69841700765077879582001822177, −6.50476380347729766275051045196, −6.26132700207302473191366763184, −5.25076559377468637918874509963, −4.19321956922146612971512865053, −3.28003340764181301950034563965, −1.72321501766681537817668643494, −0.913338161695122326305420130911,
0.913338161695122326305420130911, 1.72321501766681537817668643494, 3.28003340764181301950034563965, 4.19321956922146612971512865053, 5.25076559377468637918874509963, 6.26132700207302473191366763184, 6.50476380347729766275051045196, 7.69841700765077879582001822177, 8.901296996395076179144369887469, 9.197534866637912540350401812402
