L(s) = 1 | − 2-s + 3-s + 4-s + 1.41·5-s − 6-s − 8-s + 9-s − 1.41·10-s + 12-s + 1.41·15-s + 16-s − 1.41·17-s − 18-s + 1.41·20-s − 24-s + 1.00·25-s + 27-s − 1.41·30-s − 32-s + 1.41·34-s + 36-s − 1.41·40-s − 1.41·41-s + 1.41·43-s + 1.41·45-s + 48-s + 49-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s + 1.41·5-s − 6-s − 8-s + 9-s − 1.41·10-s + 12-s + 1.41·15-s + 16-s − 1.41·17-s − 18-s + 1.41·20-s − 24-s + 1.00·25-s + 27-s − 1.41·30-s − 32-s + 1.41·34-s + 36-s − 1.41·40-s − 1.41·41-s + 1.41·43-s + 1.41·45-s + 48-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.354874869\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.354874869\) |
\(L(1)\) |
|
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 | \( 1 - T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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
−9.153185246128506526850677113033, −8.879276186849607427647040622757, −7.987965655892556815219143801371, −7.07082369257477852149992716431, −6.46815055191435980337744499833, −5.59971734959201756791184949995, −4.36522854736731233247351960732, −3.02775246984645232156406584006, −2.25298910163709615521970108812, −1.52787103337297115316766718154,
1.52787103337297115316766718154, 2.25298910163709615521970108812, 3.02775246984645232156406584006, 4.36522854736731233247351960732, 5.59971734959201756791184949995, 6.46815055191435980337744499833, 7.07082369257477852149992716431, 7.987965655892556815219143801371, 8.879276186849607427647040622757, 9.153185246128506526850677113033