L(s) = 1 | − 2-s + 3-s + 4-s − 2.06·5-s − 6-s − 2.54·7-s − 8-s + 9-s + 2.06·10-s − 0.325·11-s + 12-s − 13-s + 2.54·14-s − 2.06·15-s + 16-s − 7.70·17-s − 18-s − 1.05·19-s − 2.06·20-s − 2.54·21-s + 0.325·22-s − 2.56·23-s − 24-s − 0.725·25-s + 26-s + 27-s − 2.54·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.924·5-s − 0.408·6-s − 0.960·7-s − 0.353·8-s + 0.333·9-s + 0.653·10-s − 0.0980·11-s + 0.288·12-s − 0.277·13-s + 0.679·14-s − 0.533·15-s + 0.250·16-s − 1.86·17-s − 0.235·18-s − 0.242·19-s − 0.462·20-s − 0.554·21-s + 0.0693·22-s − 0.535·23-s − 0.204·24-s − 0.145·25-s + 0.196·26-s + 0.192·27-s − 0.480·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4270616058\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4270616058\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 2.06T + 5T^{2} \) |
| 7 | \( 1 + 2.54T + 7T^{2} \) |
| 11 | \( 1 + 0.325T + 11T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 19 | \( 1 + 1.05T + 19T^{2} \) |
| 23 | \( 1 + 2.56T + 23T^{2} \) |
| 29 | \( 1 - 0.0777T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 5.75T + 41T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 9.54T + 73T^{2} \) |
| 79 | \( 1 + 7.67T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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
−7.925573679505603019012407397208, −7.15080997822068826981045556517, −6.79042039439879544336474031899, −6.00730191266041271521928600899, −4.92929833077216669981382201971, −3.99727595627337138094640568960, −3.53295976870014973292094170144, −2.56173120712934869954486558508, −1.87869452479561889134218754230, −0.32721110434314057628398027454,
0.32721110434314057628398027454, 1.87869452479561889134218754230, 2.56173120712934869954486558508, 3.53295976870014973292094170144, 3.99727595627337138094640568960, 4.92929833077216669981382201971, 6.00730191266041271521928600899, 6.79042039439879544336474031899, 7.15080997822068826981045556517, 7.925573679505603019012407397208