| L(s) = 1 | + 5-s − 3.41·7-s + 1.41·11-s + 4.24·13-s − 2.82·17-s − 19-s + 4.82·23-s + 25-s − 2.24·29-s − 8.82·31-s − 3.41·35-s − 7.07·37-s − 2.24·41-s − 1.75·43-s − 4.82·47-s + 4.65·49-s + 12.4·53-s + 1.41·55-s + 2.82·59-s − 8·61-s + 4.24·65-s − 11.3·67-s + 5.17·71-s − 3.65·73-s − 4.82·77-s − 2.34·79-s + 6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.29·7-s + 0.426·11-s + 1.17·13-s − 0.685·17-s − 0.229·19-s + 1.00·23-s + 0.200·25-s − 0.416·29-s − 1.58·31-s − 0.577·35-s − 1.16·37-s − 0.350·41-s − 0.267·43-s − 0.704·47-s + 0.665·49-s + 1.71·53-s + 0.190·55-s + 0.368·59-s − 1.02·61-s + 0.526·65-s − 1.38·67-s + 0.613·71-s − 0.428·73-s − 0.550·77-s − 0.263·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−7.40822281062919632087102327135, −6.81106919662421382031775568073, −6.27095459162006745402594147574, −5.66554453506068935836636116423, −4.79975953044467241390096671851, −3.68887080142984439022792364290, −3.38577645130311878586895899129, −2.27616559192727376983322793026, −1.31747480063061932985533260691, 0,
1.31747480063061932985533260691, 2.27616559192727376983322793026, 3.38577645130311878586895899129, 3.68887080142984439022792364290, 4.79975953044467241390096671851, 5.66554453506068935836636116423, 6.27095459162006745402594147574, 6.81106919662421382031775568073, 7.40822281062919632087102327135
