| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 26·5-s − 36·6-s − 49·7-s − 64·8-s + 81·9-s − 104·10-s + 664·11-s + 144·12-s + 318·13-s + 196·14-s + 234·15-s + 256·16-s + 1.58e3·17-s − 324·18-s + 236·19-s + 416·20-s − 441·21-s − 2.65e3·22-s + 2.21e3·23-s − 576·24-s − 2.44e3·25-s − 1.27e3·26-s + 729·27-s − 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.465·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.328·10-s + 1.65·11-s + 0.288·12-s + 0.521·13-s + 0.267·14-s + 0.268·15-s + 1/4·16-s + 1.32·17-s − 0.235·18-s + 0.149·19-s + 0.232·20-s − 0.218·21-s − 1.16·22-s + 0.871·23-s − 0.204·24-s − 0.783·25-s − 0.369·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.604148431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.604148431\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 26 T + p^{5} T^{2} \) |
| 11 | \( 1 - 664 T + p^{5} T^{2} \) |
| 13 | \( 1 - 318 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1582 T + p^{5} T^{2} \) |
| 19 | \( 1 - 236 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2212 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4954 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7128 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4358 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10542 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8452 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5352 T + p^{5} T^{2} \) |
| 53 | \( 1 + 33354 T + p^{5} T^{2} \) |
| 59 | \( 1 + 15436 T + p^{5} T^{2} \) |
| 61 | \( 1 + 36762 T + p^{5} T^{2} \) |
| 67 | \( 1 - 40972 T + p^{5} T^{2} \) |
| 71 | \( 1 + 9092 T + p^{5} T^{2} \) |
| 73 | \( 1 + 73454 T + p^{5} T^{2} \) |
| 79 | \( 1 - 89400 T + p^{5} T^{2} \) |
| 83 | \( 1 + 6428 T + p^{5} T^{2} \) |
| 89 | \( 1 + 122658 T + p^{5} T^{2} \) |
| 97 | \( 1 - 21370 T + p^{5} T^{2} \) |
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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
−14.96066037683515807518789254277, −14.02558213272790024076451752766, −12.62420720661657490498070096154, −11.24976535444859459911334072796, −9.702904309423403175237025968529, −9.042382273677078234612860967986, −7.48292139659386949088251555634, −6.06450826353750466648866217386, −3.52314199137743227571797404234, −1.45318512148560827288835303215,
1.45318512148560827288835303215, 3.52314199137743227571797404234, 6.06450826353750466648866217386, 7.48292139659386949088251555634, 9.042382273677078234612860967986, 9.702904309423403175237025968529, 11.24976535444859459911334072796, 12.62420720661657490498070096154, 14.02558213272790024076451752766, 14.96066037683515807518789254277
