| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 4·11-s + 2·13-s − 15-s − 6·17-s − 21-s + 25-s − 27-s + 6·29-s − 4·33-s + 35-s + 6·37-s − 2·39-s − 10·41-s + 45-s + 12·47-s + 49-s + 6·51-s + 6·53-s + 4·55-s − 4·59-s − 2·61-s + 63-s + 2·65-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 − 1.45·17-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.320·39-s − 1.56·41-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 0.539·55-s − 0.520·59-s − 0.256·61-s + 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.100521510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.100521510\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.086263978145470320871906702176, −6.93572095256684649658561321510, −6.63227079773837611228374650476, −5.96236976423631761120300661544, −5.16069304611080512499615151585, −4.38169946839211922588490194591, −3.82779934470643671119091671229, −2.60167150888032322443544236763, −1.69759352109802966237499915850, −0.807505773130075938308794163708,
0.807505773130075938308794163708, 1.69759352109802966237499915850, 2.60167150888032322443544236763, 3.82779934470643671119091671229, 4.38169946839211922588490194591, 5.16069304611080512499615151585, 5.96236976423631761120300661544, 6.63227079773837611228374650476, 6.93572095256684649658561321510, 8.086263978145470320871906702176
