| L(s) = 1 | + 5-s + 4·7-s + 11-s − 4·13-s + 6·17-s − 2·19-s − 6·23-s + 25-s + 6·29-s − 8·31-s + 4·35-s + 2·37-s + 6·41-s + 10·43-s − 6·47-s + 9·49-s − 6·53-s + 55-s + 8·61-s − 4·65-s + 4·67-s − 6·71-s + 14·73-s + 4·77-s + 16·79-s + 12·83-s + 6·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.301·11-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s + 0.937·41-s + 1.52·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s + 1.02·61-s − 0.496·65-s + 0.488·67-s − 0.712·71-s + 1.63·73-s + 0.455·77-s + 1.80·79-s + 1.31·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.814223384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.814223384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.892968827861982496277133355226, −7.34354612586582969610668722928, −6.39861240595422416599157142649, −5.64271218422640710633321581528, −5.06909587240738937197667130103, −4.42689663739240872938818776328, −3.59136555451316434677447074899, −2.42798255540264089634238069287, −1.85531615333712359450585355949, −0.861762193736076634455170442249,
0.861762193736076634455170442249, 1.85531615333712359450585355949, 2.42798255540264089634238069287, 3.59136555451316434677447074899, 4.42689663739240872938818776328, 5.06909587240738937197667130103, 5.64271218422640710633321581528, 6.39861240595422416599157142649, 7.34354612586582969610668722928, 7.892968827861982496277133355226
