| L(s) = 1 | − 5-s − 2·7-s + 5·11-s − 13-s + 17-s − 5·19-s + 23-s − 4·25-s + 2·29-s − 2·31-s + 2·35-s + 4·37-s + 5·41-s + 43-s − 3·49-s − 12·53-s − 5·55-s + 10·59-s − 10·61-s + 65-s − 4·67-s − 8·71-s + 12·73-s − 10·77-s + 6·79-s + 8·83-s − 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.50·11-s − 0.277·13-s + 0.242·17-s − 1.14·19-s + 0.208·23-s − 4/5·25-s + 0.371·29-s − 0.359·31-s + 0.338·35-s + 0.657·37-s + 0.780·41-s + 0.152·43-s − 3/7·49-s − 1.64·53-s − 0.674·55-s + 1.30·59-s − 1.28·61-s + 0.124·65-s − 0.488·67-s − 0.949·71-s + 1.40·73-s − 1.13·77-s + 0.675·79-s + 0.878·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.907623329233303902109547187224, −7.12057236198560513846405664270, −6.39863852805448877438135046128, −5.99898544413651539795295578755, −4.79098912758088521265177025201, −4.04772602345486066856015528317, −3.48222454932699347452121909471, −2.44755617113484681220187205717, −1.31055766523076933566859515058, 0,
1.31055766523076933566859515058, 2.44755617113484681220187205717, 3.48222454932699347452121909471, 4.04772602345486066856015528317, 4.79098912758088521265177025201, 5.99898544413651539795295578755, 6.39863852805448877438135046128, 7.12057236198560513846405664270, 7.907623329233303902109547187224
