| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 13-s − 15-s − 2·19-s − 4·21-s + 25-s − 27-s + 2·29-s + 4·35-s − 6·37-s − 39-s + 6·41-s + 45-s + 9·49-s − 10·53-s + 2·57-s − 12·59-s + 4·63-s + 65-s − 12·67-s + 12·71-s + 6·73-s − 75-s + 10·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 0.458·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.676·35-s − 0.986·37-s − 0.160·39-s + 0.937·41-s + 0.149·45-s + 9/7·49-s − 1.37·53-s + 0.264·57-s − 1.56·59-s + 0.503·63-s + 0.124·65-s − 1.46·67-s + 1.42·71-s + 0.702·73-s − 0.115·75-s + 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.857584002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.857584002\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.16653742776209, −12.50828453463264, −12.10843487292906, −11.79174365728056, −11.02770044914891, −10.82467799693357, −10.59207146892928, −9.834462998247805, −9.281430705403967, −8.905377599391583, −8.239879298741479, −7.858797296949835, −7.478365320054960, −6.612580163992547, −6.436382120275255, −5.687909810936467, −5.324067127437890, −4.676349741695251, −4.509984266135425, −3.724718221819976, −3.061410540173632, −2.243933931420480, −1.768295824781353, −1.263271135969160, −0.5146808537150006,
0.5146808537150006, 1.263271135969160, 1.768295824781353, 2.243933931420480, 3.061410540173632, 3.724718221819976, 4.509984266135425, 4.676349741695251, 5.324067127437890, 5.687909810936467, 6.436382120275255, 6.612580163992547, 7.478365320054960, 7.858797296949835, 8.239879298741479, 8.905377599391583, 9.281430705403967, 9.834462998247805, 10.59207146892928, 10.82467799693357, 11.02770044914891, 11.79174365728056, 12.10843487292906, 12.50828453463264, 13.16653742776209
