L(s) = 1 | + 2-s − 3-s + 4-s + 3.82·5-s − 6-s + 8-s + 9-s + 3.82·10-s − 5.10·11-s − 12-s − 0.0556·13-s − 3.82·15-s + 16-s + 6.77·17-s + 18-s − 4.16·19-s + 3.82·20-s − 5.10·22-s + 23-s − 24-s + 9.65·25-s − 0.0556·26-s − 27-s + 2.94·29-s − 3.82·30-s + 3.21·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.71·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.21·10-s − 1.54·11-s − 0.288·12-s − 0.0154·13-s − 0.988·15-s + 0.250·16-s + 1.64·17-s + 0.235·18-s − 0.955·19-s + 0.856·20-s − 1.08·22-s + 0.208·23-s − 0.204·24-s + 1.93·25-s − 0.0109·26-s − 0.192·27-s + 0.546·29-s − 0.698·30-s + 0.578·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.704942964\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.704942964\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 + 0.0556T + 13T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 + 4.16T + 19T^{2} \) |
| 29 | \( 1 - 2.94T + 29T^{2} \) |
| 31 | \( 1 - 3.21T + 31T^{2} \) |
| 37 | \( 1 + 8.05T + 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 5.71T + 53T^{2} \) |
| 59 | \( 1 + 8.60T + 59T^{2} \) |
| 61 | \( 1 - 5.77T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 2.21T + 73T^{2} \) |
| 79 | \( 1 + 2.94T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 2.39T + 89T^{2} \) |
| 97 | \( 1 + 4.60T + 97T^{2} \) |
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
−7.85025332480923657915007959799, −7.04992586792725297121736948190, −6.24443130596160719600837196049, −5.82273815703738508156348502165, −5.16585903080697192055642074753, −4.81564802158585044038272309813, −3.53100962178339838323815095959, −2.62223104914888469049735687188, −2.03259966197757748482384965238, −0.929069250681380349996872931449,
0.929069250681380349996872931449, 2.03259966197757748482384965238, 2.62223104914888469049735687188, 3.53100962178339838323815095959, 4.81564802158585044038272309813, 5.16585903080697192055642074753, 5.82273815703738508156348502165, 6.24443130596160719600837196049, 7.04992586792725297121736948190, 7.85025332480923657915007959799