L(s) = 1 | − 2-s − 3-s + 4-s − 0.0775·5-s + 6-s + 2.74·7-s − 8-s + 9-s + 0.0775·10-s − 11-s − 12-s + 4.76·13-s − 2.74·14-s + 0.0775·15-s + 16-s + 4.60·17-s − 18-s − 5.24·19-s − 0.0775·20-s − 2.74·21-s + 22-s − 9.37·23-s + 24-s − 4.99·25-s − 4.76·26-s − 27-s + 2.74·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0346·5-s + 0.408·6-s + 1.03·7-s − 0.353·8-s + 0.333·9-s + 0.0245·10-s − 0.301·11-s − 0.288·12-s + 1.32·13-s − 0.732·14-s + 0.0200·15-s + 0.250·16-s + 1.11·17-s − 0.235·18-s − 1.20·19-s − 0.0173·20-s − 0.598·21-s + 0.213·22-s − 1.95·23-s + 0.204·24-s − 0.998·25-s − 0.935·26-s − 0.192·27-s + 0.518·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 0.0775T + 5T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 13 | \( 1 - 4.76T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 + 5.24T + 19T^{2} \) |
| 23 | \( 1 + 9.37T + 23T^{2} \) |
| 29 | \( 1 + 4.01T + 29T^{2} \) |
| 31 | \( 1 - 3.80T + 31T^{2} \) |
| 37 | \( 1 + 0.483T + 37T^{2} \) |
| 41 | \( 1 - 1.38T + 41T^{2} \) |
| 43 | \( 1 + 5.58T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 4.36T + 53T^{2} \) |
| 59 | \( 1 + 4.92T + 59T^{2} \) |
| 67 | \( 1 + 2.49T + 67T^{2} \) |
| 71 | \( 1 - 0.600T + 71T^{2} \) |
| 73 | \( 1 - 8.32T + 73T^{2} \) |
| 79 | \( 1 + 0.850T + 79T^{2} \) |
| 83 | \( 1 + 0.613T + 83T^{2} \) |
| 89 | \( 1 - 0.357T + 89T^{2} \) |
| 97 | \( 1 - 9.29T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.023130013832405490500967322582, −7.71830792528675282505164142474, −6.44057474464022045062371343668, −6.05041555096889110910775842750, −5.22835227253193985607882928780, −4.25837770274494831191350596390, −3.47691897808482479205335829544, −2.02006765030001935916242400194, −1.41325197757227575605254124140, 0,
1.41325197757227575605254124140, 2.02006765030001935916242400194, 3.47691897808482479205335829544, 4.25837770274494831191350596390, 5.22835227253193985607882928780, 6.05041555096889110910775842750, 6.44057474464022045062371343668, 7.71830792528675282505164142474, 8.023130013832405490500967322582