| L(s) = 1 | + 4.05·2-s + 8.44·4-s + 9.92·5-s + 1.80·8-s + 40.2·10-s + 13.5·11-s + 18.5·13-s − 60.2·16-s + 93.7·17-s + 131.·19-s + 83.7·20-s + 54.9·22-s + 198.·23-s − 26.5·25-s + 75.2·26-s − 188.·29-s − 83.9·31-s − 258.·32-s + 380.·34-s + 80.1·37-s + 534.·38-s + 17.9·40-s + 385.·41-s − 397.·43-s + 114.·44-s + 804.·46-s − 272.·47-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.05·4-s + 0.887·5-s + 0.0799·8-s + 1.27·10-s + 0.371·11-s + 0.395·13-s − 0.941·16-s + 1.33·17-s + 1.59·19-s + 0.936·20-s + 0.532·22-s + 1.79·23-s − 0.212·25-s + 0.567·26-s − 1.20·29-s − 0.486·31-s − 1.42·32-s + 1.91·34-s + 0.356·37-s + 2.28·38-s + 0.0709·40-s + 1.46·41-s − 1.40·43-s + 0.391·44-s + 2.57·46-s − 0.845·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.122744448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.122744448\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.05T + 8T^{2} \) |
| 5 | \( 1 - 9.92T + 125T^{2} \) |
| 11 | \( 1 - 13.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 198.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 83.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 80.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 385.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 397.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 36.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 395.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 13.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 340.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 211.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 486.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 293.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 889.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−11.05735149500400698717630791293, −9.721830433457583285190277804630, −9.173722847259809136181556697300, −7.65198067657847716597994820688, −6.61655896011836958560343290966, −5.60071182589537325431796709793, −5.17196766508970463786798420243, −3.73878281905455488927684908677, −2.90433292223298311564753552400, −1.36650983530787149558143635750,
1.36650983530787149558143635750, 2.90433292223298311564753552400, 3.73878281905455488927684908677, 5.17196766508970463786798420243, 5.60071182589537325431796709793, 6.61655896011836958560343290966, 7.65198067657847716597994820688, 9.173722847259809136181556697300, 9.721830433457583285190277804630, 11.05735149500400698717630791293
