| L(s) = 1 | + 2·5-s − 2·13-s − 6·17-s − 19-s + 2·23-s − 25-s + 6·29-s + 31-s + 2·37-s + 2·41-s + 9·43-s − 2·47-s − 6·53-s − 8·59-s + 11·61-s − 4·65-s − 12·67-s + 4·71-s + 5·73-s − 4·79-s + 4·83-s − 12·85-s + 18·89-s − 2·95-s + 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.554·13-s − 1.45·17-s − 0.229·19-s + 0.417·23-s − 1/5·25-s + 1.11·29-s + 0.179·31-s + 0.328·37-s + 0.312·41-s + 1.37·43-s − 0.291·47-s − 0.824·53-s − 1.04·59-s + 1.40·61-s − 0.496·65-s − 1.46·67-s + 0.474·71-s + 0.585·73-s − 0.450·79-s + 0.439·83-s − 1.30·85-s + 1.90·89-s − 0.205·95-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.186954936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.186954936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−16.63506540276470, −15.85801744645869, −15.55800820902900, −14.70220148953648, −14.32812229662163, −13.53603491152153, −13.31444499609977, −12.53914371538429, −12.04892088590418, −11.17207402097078, −10.78614596429740, −10.05235148842123, −9.503026576363068, −8.983192457266467, −8.332254632809977, −7.544864296026101, −6.836933354946556, −6.246370862458727, −5.705923895831418, −4.731074153788725, −4.417950193372267, −3.252091073264144, −2.425365315517188, −1.888699069535794, −0.6756914934075974,
0.6756914934075974, 1.888699069535794, 2.425365315517188, 3.252091073264144, 4.417950193372267, 4.731074153788725, 5.705923895831418, 6.246370862458727, 6.836933354946556, 7.544864296026101, 8.332254632809977, 8.983192457266467, 9.503026576363068, 10.05235148842123, 10.78614596429740, 11.17207402097078, 12.04892088590418, 12.53914371538429, 13.31444499609977, 13.53603491152153, 14.32812229662163, 14.70220148953648, 15.55800820902900, 15.85801744645869, 16.63506540276470
