| L(s) = 1 | − 3-s − 2·9-s − 3·11-s − 3·13-s − 3·17-s + 2·19-s + 23-s + 5·27-s + 3·29-s + 4·31-s + 3·33-s + 8·37-s + 3·39-s + 10·41-s + 4·43-s − 13·47-s + 3·51-s − 2·57-s + 8·59-s + 10·61-s − 6·67-s − 69-s − 6·73-s + 7·79-s + 81-s − 3·87-s + 10·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.904·11-s − 0.832·13-s − 0.727·17-s + 0.458·19-s + 0.208·23-s + 0.962·27-s + 0.557·29-s + 0.718·31-s + 0.522·33-s + 1.31·37-s + 0.480·39-s + 1.56·41-s + 0.609·43-s − 1.89·47-s + 0.420·51-s − 0.264·57-s + 1.04·59-s + 1.28·61-s − 0.733·67-s − 0.120·69-s − 0.702·73-s + 0.787·79-s + 1/9·81-s − 0.321·87-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.554505253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.554505253\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−12.97645666301751, −12.44560429527284, −11.97736256407923, −11.50385674044111, −11.12558771345301, −10.77206054291990, −10.06776743467469, −9.855384274185082, −9.206740481740942, −8.735608875291930, −8.117414809010163, −7.818117870406447, −7.243043439680432, −6.649026740041630, −6.240759856910366, −5.661180259793588, −5.267502563512882, −4.650158126251347, −4.432639937799156, −3.528119397642373, −2.749392557172097, −2.641412182994381, −1.904216344030614, −0.8501567097184407, −0.4616243050448906,
0.4616243050448906, 0.8501567097184407, 1.904216344030614, 2.641412182994381, 2.749392557172097, 3.528119397642373, 4.432639937799156, 4.650158126251347, 5.267502563512882, 5.661180259793588, 6.240759856910366, 6.649026740041630, 7.243043439680432, 7.818117870406447, 8.117414809010163, 8.735608875291930, 9.206740481740942, 9.855384274185082, 10.06776743467469, 10.77206054291990, 11.12558771345301, 11.50385674044111, 11.97736256407923, 12.44560429527284, 12.97645666301751
