| L(s) = 1 | − 3·3-s + 15.2·5-s + 7·7-s + 9·9-s − 26.3·11-s + 73.8·13-s − 45.6·15-s + 7.69·17-s − 69.3·19-s − 21·21-s + 74.6·23-s + 106.·25-s − 27·27-s + 145.·29-s + 79.2·31-s + 78.9·33-s + 106.·35-s + 5.82·37-s − 221.·39-s + 203.·41-s + 95.6·43-s + 137.·45-s + 471.·47-s + 49·49-s − 23.0·51-s − 361.·53-s − 400.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.36·5-s + 0.377·7-s + 0.333·9-s − 0.720·11-s + 1.57·13-s − 0.786·15-s + 0.109·17-s − 0.837·19-s − 0.218·21-s + 0.676·23-s + 0.854·25-s − 0.192·27-s + 0.931·29-s + 0.459·31-s + 0.416·33-s + 0.514·35-s + 0.0259·37-s − 0.909·39-s + 0.774·41-s + 0.339·43-s + 0.453·45-s + 1.46·47-s + 0.142·49-s − 0.0634·51-s − 0.936·53-s − 0.981·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.653789076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.653789076\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 7T \) |
| good | 5 | \( 1 - 15.2T + 125T^{2} \) |
| 11 | \( 1 + 26.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 73.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.69T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 74.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 79.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.82T + 5.06e4T^{2} \) |
| 41 | \( 1 - 203.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 95.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 471.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 361.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 834.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 734.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 624.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 202.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 830.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 848.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 778.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 400.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 119.T + 9.12e5T^{2} \) |
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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
−9.243855135820888300190604468026, −8.562658676722154296297574494085, −7.60921433711674256541015613819, −6.38864260578475342474464262680, −6.04640333793175955032166746280, −5.19032669859092107079004437724, −4.30519222575532088163390635248, −2.92037590904263606967820347813, −1.83363086504664927059073372871, −0.884914300537558433883225134824,
0.884914300537558433883225134824, 1.83363086504664927059073372871, 2.92037590904263606967820347813, 4.30519222575532088163390635248, 5.19032669859092107079004437724, 6.04640333793175955032166746280, 6.38864260578475342474464262680, 7.60921433711674256541015613819, 8.562658676722154296297574494085, 9.243855135820888300190604468026
