L(s) = 1 | − 4.12·2-s − 3·3-s + 9·4-s + 13.4·5-s + 12.3·6-s − 31.4·7-s − 4.12·8-s + 9·9-s − 55.4·10-s + 40.4·11-s − 27·12-s + 129.·14-s − 40.3·15-s − 55·16-s − 43.1·17-s − 37.1·18-s − 26.9·19-s + 120.·20-s + 94.2·21-s − 166.·22-s − 19.0·23-s + 12.3·24-s + 55.6·25-s − 27·27-s − 282.·28-s − 154.·29-s + 166.·30-s + ⋯ |
L(s) = 1 | − 1.45·2-s − 0.577·3-s + 1.12·4-s + 1.20·5-s + 0.841·6-s − 1.69·7-s − 0.182·8-s + 0.333·9-s − 1.75·10-s + 1.11·11-s − 0.649·12-s + 2.47·14-s − 0.694·15-s − 0.859·16-s − 0.615·17-s − 0.485·18-s − 0.325·19-s + 1.35·20-s + 0.979·21-s − 1.61·22-s − 0.172·23-s + 0.105·24-s + 0.445·25-s − 0.192·27-s − 1.90·28-s − 0.986·29-s + 1.01·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 4.12T + 8T^{2} \) |
| 5 | \( 1 - 13.4T + 125T^{2} \) |
| 7 | \( 1 + 31.4T + 343T^{2} \) |
| 11 | \( 1 - 40.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 43.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 308.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 43.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 47.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 342.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 133.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 590.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 541.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 230.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 389.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 897.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 925.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.56e3T + 9.12e5T^{2} \) |
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
−9.755624259469251111108886627608, −9.480196045044593693961056231165, −8.655896913034123005434833329213, −7.18865853427112873591183271993, −6.44052118598353183017102799416, −5.93187246775876670481689550050, −4.20841270190841640600250909533, −2.55472301057076868574984331212, −1.27269060929327938250794938671, 0,
1.27269060929327938250794938671, 2.55472301057076868574984331212, 4.20841270190841640600250909533, 5.93187246775876670481689550050, 6.44052118598353183017102799416, 7.18865853427112873591183271993, 8.655896913034123005434833329213, 9.480196045044593693961056231165, 9.755624259469251111108886627608