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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.366796737\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.366796737\) |
\(L(1)\) |
|
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 | \( 1 + 2T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.536829639078312526742371409379, −8.184645454819341045405136736520, −7.46807052328453119794748255448, −7.04374713855224369388064666607, −5.90570503240493090306770099654, −5.40200547686342284004699658787, −4.28647837715319173215694718811, −3.94448536080837266151985552466, −2.80310220339979786929606683550, −1.11827589940167191063800938637,
1.11827589940167191063800938637, 2.80310220339979786929606683550, 3.94448536080837266151985552466, 4.28647837715319173215694718811, 5.40200547686342284004699658787, 5.90570503240493090306770099654, 7.04374713855224369388064666607, 7.46807052328453119794748255448, 8.184645454819341045405136736520, 9.536829639078312526742371409379