L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 6·11-s − 2·13-s − 15-s + 4·19-s + 2·21-s + 23-s + 25-s + 27-s − 2·29-s + 8·31-s − 6·33-s − 2·35-s − 4·37-s − 2·39-s + 2·41-s + 8·43-s − 45-s − 3·49-s − 2·53-s + 6·55-s + 4·57-s + 4·59-s + 2·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.258·15-s + 0.917·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 1.04·33-s − 0.338·35-s − 0.657·37-s − 0.320·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s − 3/7·49-s − 0.274·53-s + 0.809·55-s + 0.529·57-s + 0.520·59-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.087939769\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.087939769\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−8.051863093541795598975249791605, −7.62065055953918860500991927099, −7.05310930657121520458435489780, −5.88977276110056709889840142473, −5.01541577404560810095238487752, −4.69413082171999456728781930223, −3.55949717461122418634756189946, −2.77161363830144351426460465778, −2.08481145506420277964800270433, −0.73909227737322239626033971422,
0.73909227737322239626033971422, 2.08481145506420277964800270433, 2.77161363830144351426460465778, 3.55949717461122418634756189946, 4.69413082171999456728781930223, 5.01541577404560810095238487752, 5.88977276110056709889840142473, 7.05310930657121520458435489780, 7.62065055953918860500991927099, 8.051863093541795598975249791605