L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 4·11-s − 2·13-s − 15-s − 2·17-s − 4·19-s − 21-s + 23-s + 25-s + 27-s − 2·29-s − 4·33-s + 35-s − 6·37-s − 2·39-s − 6·41-s − 8·43-s − 45-s − 4·47-s + 49-s − 2·51-s − 2·53-s + 4·55-s − 4·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.169·35-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.66798944460280, −13.18543500740142, −13.02058409919808, −12.39905373601147, −12.03681965235792, −11.26278949256080, −11.02544300325395, −10.23474386495665, −10.07551348393111, −9.556539823511811, −8.772401279570631, −8.460157471141318, −8.146964245303965, −7.417564187886179, −7.020018152951262, −6.649391009106429, −5.887749546908241, −5.247539793358231, −4.827699175144347, −4.274868865068841, −3.553680753078629, −3.208974622467966, −2.431201053583410, −2.128964427268762, −1.263228026015961, 0, 0,
1.263228026015961, 2.128964427268762, 2.431201053583410, 3.208974622467966, 3.553680753078629, 4.274868865068841, 4.827699175144347, 5.247539793358231, 5.887749546908241, 6.649391009106429, 7.020018152951262, 7.417564187886179, 8.146964245303965, 8.460157471141318, 8.772401279570631, 9.556539823511811, 10.07551348393111, 10.23474386495665, 11.02544300325395, 11.26278949256080, 12.03681965235792, 12.39905373601147, 13.02058409919808, 13.18543500740142, 13.66798944460280