| L(s) = 1 | + 2-s − 3-s + 4-s + 2.70·5-s − 6-s + 7-s + 8-s + 9-s + 2.70·10-s − 0.701·11-s − 12-s + 13-s + 14-s − 2.70·15-s + 16-s − 2.70·17-s + 18-s − 0.701·19-s + 2.70·20-s − 21-s − 0.701·22-s + 4.70·23-s − 24-s + 2.29·25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.20·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.854·10-s − 0.211·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.697·15-s + 0.250·16-s − 0.655·17-s + 0.235·18-s − 0.160·19-s + 0.604·20-s − 0.218·21-s − 0.149·22-s + 0.980·23-s − 0.204·24-s + 0.459·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.318810183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.318810183\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.70T + 5T^{2} \) |
| 11 | \( 1 + 0.701T + 11T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + 0.701T + 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 - 2.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 - 1.29T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 8.80T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−10.95902301042485306900855631394, −10.09375930711130694623145937064, −9.220752912421352144473568592103, −8.021259355367973261734314365488, −6.77192914266805202384889861604, −6.11387749439743136317593171945, −5.23069746258966728509319936006, −4.41834128660271042071707525915, −2.81767171958026255079962866204, −1.56033051386842278131548780488,
1.56033051386842278131548780488, 2.81767171958026255079962866204, 4.41834128660271042071707525915, 5.23069746258966728509319936006, 6.11387749439743136317593171945, 6.77192914266805202384889861604, 8.021259355367973261734314365488, 9.220752912421352144473568592103, 10.09375930711130694623145937064, 10.95902301042485306900855631394
