| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 2·11-s + 2·13-s − 15-s − 21-s + 4·23-s − 4·25-s + 27-s − 6·29-s − 5·31-s − 2·33-s + 35-s − 37-s + 2·39-s + 2·41-s − 45-s − 47-s − 6·49-s + 9·53-s + 2·55-s − 5·59-s − 12·61-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 − 0.603·11-s + 0.554·13-s − 0.258·15-s − 0.218·21-s + 0.834·23-s − 4/5·25-s + 0.192·27-s − 1.11·29-s − 0.898·31-s − 0.348·33-s + 0.169·35-s − 0.164·37-s + 0.320·39-s + 0.312·41-s − 0.149·45-s − 0.145·47-s − 6/7·49-s + 1.23·53-s + 0.269·55-s − 0.650·59-s − 1.53·61-s − 0.125·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.945106184052585139758379575266, −7.55308257744982315753758664140, −6.73311984688068827108858761641, −5.85129300030107946915719638810, −5.07098989305488583971338291559, −4.04530078291200321778071830236, −3.45053522856135647539257962566, −2.59211203970254782164506873562, −1.50704790034368395444312726686, 0,
1.50704790034368395444312726686, 2.59211203970254782164506873562, 3.45053522856135647539257962566, 4.04530078291200321778071830236, 5.07098989305488583971338291559, 5.85129300030107946915719638810, 6.73311984688068827108858761641, 7.55308257744982315753758664140, 7.945106184052585139758379575266
