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 − 4·11-s + 12-s − 0.701·13-s − 14-s − 2.70·15-s + 16-s + 4·17-s − 18-s + 7.40·19-s − 2.70·20-s + 21-s + 4·22-s + 23-s − 24-s + 2.29·25-s + 0.701·26-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 − 1.20·11-s + 0.288·12-s − 0.194·13-s − 0.267·14-s − 0.697·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s + 1.69·19-s − 0.604·20-s + 0.218·21-s + 0.852·22-s + 0.208·23-s − 0.204·24-s + 0.459·25-s + 0.137·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.128020792\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128020792\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 0.701T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 7.40T + 19T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 - 5.40T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 0.596T + 83T^{2} \) |
| 89 | \( 1 - 1.40T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−10.00426136986052802384022504771, −9.095606485511218683429208318697, −8.125110480959164981723335819844, −7.71561780347726697298713687871, −7.20344562908946100144309530119, −5.68124938463567348277028064385, −4.67064693646732327554151233917, −3.44595235509003519125432913664, −2.62180933416597997887301070516, −0.917304471445257710921366129008,
0.917304471445257710921366129008, 2.62180933416597997887301070516, 3.44595235509003519125432913664, 4.67064693646732327554151233917, 5.68124938463567348277028064385, 7.20344562908946100144309530119, 7.71561780347726697298713687871, 8.125110480959164981723335819844, 9.095606485511218683429208318697, 10.00426136986052802384022504771