| L(s) = 1 | − 5-s − 5·11-s + 13-s + 17-s − 6·19-s − 3·23-s + 25-s + 6·29-s − 4·31-s − 11·37-s + 9·41-s + 6·43-s − 4·47-s + 6·53-s + 5·55-s − 9·59-s − 10·61-s − 65-s − 3·67-s − 8·71-s + 73-s + 15·79-s − 8·83-s − 85-s − 89-s + 6·95-s − 5·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.50·11-s + 0.277·13-s + 0.242·17-s − 1.37·19-s − 0.625·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 1.80·37-s + 1.40·41-s + 0.914·43-s − 0.583·47-s + 0.824·53-s + 0.674·55-s − 1.17·59-s − 1.28·61-s − 0.124·65-s − 0.366·67-s − 0.949·71-s + 0.117·73-s + 1.68·79-s − 0.878·83-s − 0.108·85-s − 0.105·89-s + 0.615·95-s − 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.06332671700308, −12.67054053486202, −12.24789424664369, −11.96575283712541, −11.11951584432555, −10.79135033559115, −10.43692722771860, −10.16652123047661, −9.196123772212038, −9.078581479522984, −8.232533135193680, −8.070775178426815, −7.633280207012088, −6.946460665494430, −6.591724838558293, −5.807427075454585, −5.582521850377178, −4.897969917092591, −4.316405648233406, −4.026101135195953, −3.127201824904379, −2.848331866985335, −2.110380440520705, −1.588260918321288, −0.5943222775978844, 0,
0.5943222775978844, 1.588260918321288, 2.110380440520705, 2.848331866985335, 3.127201824904379, 4.026101135195953, 4.316405648233406, 4.897969917092591, 5.582521850377178, 5.807427075454585, 6.591724838558293, 6.946460665494430, 7.633280207012088, 8.070775178426815, 8.232533135193680, 9.078581479522984, 9.196123772212038, 10.16652123047661, 10.43692722771860, 10.79135033559115, 11.11951584432555, 11.96575283712541, 12.24789424664369, 12.67054053486202, 13.06332671700308
