| L(s) = 1 | + 2·3-s + 9-s − 11-s + 4·17-s + 4·19-s + 6·23-s − 4·27-s + 2·29-s − 2·33-s + 6·37-s − 10·41-s + 4·43-s + 10·47-s − 7·49-s + 8·51-s − 2·53-s + 8·57-s + 4·59-s − 14·61-s + 2·67-s + 12·69-s − 4·71-s + 4·73-s + 8·79-s − 11·81-s + 12·83-s + 4·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.970·17-s + 0.917·19-s + 1.25·23-s − 0.769·27-s + 0.371·29-s − 0.348·33-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.45·47-s − 49-s + 1.12·51-s − 0.274·53-s + 1.05·57-s + 0.520·59-s − 1.79·61-s + 0.244·67-s + 1.44·69-s − 0.474·71-s + 0.468·73-s + 0.900·79-s − 1.22·81-s + 1.31·83-s + 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.064861696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.064861696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.325212115564585801398263930760, −7.71074312582419540470010210279, −7.21038339610917052837836243397, −6.17266190663608241027219380630, −5.34822899839052528837040257832, −4.59075067821992159614219087971, −3.41124306819269463835862582161, −3.09250818713578792941273196838, −2.12108049148782043580435271644, −0.958996551386610933544583929410,
0.958996551386610933544583929410, 2.12108049148782043580435271644, 3.09250818713578792941273196838, 3.41124306819269463835862582161, 4.59075067821992159614219087971, 5.34822899839052528837040257832, 6.17266190663608241027219380630, 7.21038339610917052837836243397, 7.71074312582419540470010210279, 8.325212115564585801398263930760
