| L(s) = 1 | − 2·3-s − 7-s + 9-s + 2·11-s + 4·13-s + 6·17-s + 2·21-s + 23-s + 4·27-s − 2·29-s − 4·31-s − 4·33-s − 8·39-s + 6·41-s + 6·43-s + 49-s − 12·51-s + 12·53-s + 10·59-s + 2·61-s − 63-s − 2·67-s − 2·69-s − 8·71-s − 2·73-s − 2·77-s − 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 1.45·17-s + 0.436·21-s + 0.208·23-s + 0.769·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s − 1.28·39-s + 0.937·41-s + 0.914·43-s + 1/7·49-s − 1.68·51-s + 1.64·53-s + 1.30·59-s + 0.256·61-s − 0.125·63-s − 0.244·67-s − 0.240·69-s − 0.949·71-s − 0.234·73-s − 0.227·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−14.59550846662739, −13.99964355180809, −13.34787348619123, −12.91685769285066, −12.35465315671354, −11.92845429189670, −11.48072654576097, −10.99824863025474, −10.53756801714541, −10.02449146791630, −9.445918583838110, −8.843353885073297, −8.416547294686482, −7.612265333150334, −7.112204717672683, −6.573961647989874, −5.882797758114137, −5.686762178597228, −5.211210878274085, −4.211500474407828, −3.885635004564918, −3.166995336783426, −2.450550856354063, −1.286958607683294, −1.008022617167871, 0,
1.008022617167871, 1.286958607683294, 2.450550856354063, 3.166995336783426, 3.885635004564918, 4.211500474407828, 5.211210878274085, 5.686762178597228, 5.882797758114137, 6.573961647989874, 7.112204717672683, 7.612265333150334, 8.416547294686482, 8.843353885073297, 9.445918583838110, 10.02449146791630, 10.53756801714541, 10.99824863025474, 11.48072654576097, 11.92845429189670, 12.35465315671354, 12.91685769285066, 13.34787348619123, 13.99964355180809, 14.59550846662739
