| L(s) = 1 | + 5-s + 3·7-s − 3·11-s − 13-s + 3·17-s + 4·19-s − 3·23-s + 25-s − 10·29-s + 6·31-s + 3·35-s + 5·37-s − 5·41-s + 2·43-s + 2·47-s + 2·49-s − 11·53-s − 3·55-s − 4·59-s − 61-s − 65-s − 4·67-s + 3·71-s + 6·73-s − 9·77-s − 3·79-s − 16·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.917·19-s − 0.625·23-s + 1/5·25-s − 1.85·29-s + 1.07·31-s + 0.507·35-s + 0.821·37-s − 0.780·41-s + 0.304·43-s + 0.291·47-s + 2/7·49-s − 1.51·53-s − 0.404·55-s − 0.520·59-s − 0.128·61-s − 0.124·65-s − 0.488·67-s + 0.356·71-s + 0.702·73-s − 1.02·77-s − 0.337·79-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−15.15735340493129, −14.51979149958629, −13.99710518917555, −13.80756075918349, −12.92073660991386, −12.67571012063661, −11.85354786443249, −11.46917202065086, −10.96395013703094, −10.36377715446421, −9.746564220321842, −9.491876637326872, −8.579800724264048, −8.093193538348072, −7.582678702846294, −7.268193317015630, −6.229561289215648, −5.747963995194509, −5.158012359681707, −4.771907422851499, −3.983618685021442, −3.164471868833783, −2.526578483532448, −1.765951819711093, −1.176166924497127, 0,
1.176166924497127, 1.765951819711093, 2.526578483532448, 3.164471868833783, 3.983618685021442, 4.771907422851499, 5.158012359681707, 5.747963995194509, 6.229561289215648, 7.268193317015630, 7.582678702846294, 8.093193538348072, 8.579800724264048, 9.491876637326872, 9.746564220321842, 10.36377715446421, 10.96395013703094, 11.46917202065086, 11.85354786443249, 12.67571012063661, 12.92073660991386, 13.80756075918349, 13.99710518917555, 14.51979149958629, 15.15735340493129
