L(s) = 1 | − 2-s + 3-s + 4-s + 3.52·5-s − 6-s − 2.59·7-s − 8-s + 9-s − 3.52·10-s + 5.05·11-s + 12-s + 3.52·13-s + 2.59·14-s + 3.52·15-s + 16-s − 17-s − 18-s + 1.14·19-s + 3.52·20-s − 2.59·21-s − 5.05·22-s + 5.21·23-s − 24-s + 7.42·25-s − 3.52·26-s + 27-s − 2.59·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.57·5-s − 0.408·6-s − 0.979·7-s − 0.353·8-s + 0.333·9-s − 1.11·10-s + 1.52·11-s + 0.288·12-s + 0.977·13-s + 0.692·14-s + 0.910·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.263·19-s + 0.788·20-s − 0.565·21-s − 1.07·22-s + 1.08·23-s − 0.204·24-s + 1.48·25-s − 0.691·26-s + 0.192·27-s − 0.489·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.777530051\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.777530051\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 19 | \( 1 - 1.14T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 9.13T + 37T^{2} \) |
| 41 | \( 1 - 2.09T + 41T^{2} \) |
| 43 | \( 1 - 6.42T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 9.44T + 53T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 + 7.13T + 71T^{2} \) |
| 73 | \( 1 - 7.90T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 - 8.46T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 9T + 97T^{2} \) |
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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
−8.288463230999148352275839031607, −7.31099093477463944497032807648, −6.49316648140623410775569665947, −6.29730636072305323120403033506, −5.52212625568954328867684554235, −4.28140246732743731295936564247, −3.37580613013195145341120026160, −2.68140200396602120716902944445, −1.69064151606323225980746350638, −1.04313554714109963876014217662,
1.04313554714109963876014217662, 1.69064151606323225980746350638, 2.68140200396602120716902944445, 3.37580613013195145341120026160, 4.28140246732743731295936564247, 5.52212625568954328867684554235, 6.29730636072305323120403033506, 6.49316648140623410775569665947, 7.31099093477463944497032807648, 8.288463230999148352275839031607