| L(s) = 1 | − 3-s + 3·5-s + 7-s + 9-s − 13-s − 3·15-s − 5·17-s + 4·19-s − 21-s − 4·23-s + 4·25-s − 27-s + 7·29-s − 4·31-s + 3·35-s − 5·37-s + 39-s − 9·41-s + 3·45-s + 49-s + 5·51-s + 3·53-s − 4·57-s − 2·61-s + 63-s − 3·65-s + 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s − 0.774·15-s − 1.21·17-s + 0.917·19-s − 0.218·21-s − 0.834·23-s + 4/5·25-s − 0.192·27-s + 1.29·29-s − 0.718·31-s + 0.507·35-s − 0.821·37-s + 0.160·39-s − 1.40·41-s + 0.447·45-s + 1/7·49-s + 0.700·51-s + 0.412·53-s − 0.529·57-s − 0.256·61-s + 0.125·63-s − 0.372·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.90272117072144, −15.48603316903651, −14.78348792331940, −14.04133775917367, −13.81976616634122, −13.28115904928109, −12.65672181026867, −11.99431348341795, −11.60639720840345, −10.82566605614256, −10.39968311753528, −9.821593903591757, −9.399023875234489, −8.642524113602024, −8.145767363967883, −7.121571356801314, −6.813139205029532, −6.070194982844827, −5.556340829301313, −4.999391926830294, −4.424563392192216, −3.492712257770869, −2.531828119324251, −1.915545181720184, −1.232053141036652, 0,
1.232053141036652, 1.915545181720184, 2.531828119324251, 3.492712257770869, 4.424563392192216, 4.999391926830294, 5.556340829301313, 6.070194982844827, 6.813139205029532, 7.121571356801314, 8.145767363967883, 8.642524113602024, 9.399023875234489, 9.821593903591757, 10.39968311753528, 10.82566605614256, 11.60639720840345, 11.99431348341795, 12.65672181026867, 13.28115904928109, 13.81976616634122, 14.04133775917367, 14.78348792331940, 15.48603316903651, 15.90272117072144
