| L(s) = 1 | − 0.874·5-s − 7-s − 2.82·11-s + 1.23·13-s + 3.70·17-s + 19-s + 2.82·23-s − 4.23·25-s + 7.19·29-s − 5.70·31-s + 0.874·35-s − 4.47·37-s − 8.48·41-s − 0.763·43-s + 5.45·47-s + 49-s − 4.37·53-s + 2.47·55-s + 14.8·59-s − 12.4·61-s − 1.08·65-s + 11.4·67-s + 3.03·71-s + 6.94·73-s + 2.82·77-s − 1.52·79-s − 4.78·83-s + ⋯ |
| L(s) = 1 | − 0.390·5-s − 0.377·7-s − 0.852·11-s + 0.342·13-s + 0.897·17-s + 0.229·19-s + 0.589·23-s − 0.847·25-s + 1.33·29-s − 1.02·31-s + 0.147·35-s − 0.735·37-s − 1.32·41-s − 0.116·43-s + 0.795·47-s + 0.142·49-s − 0.600·53-s + 0.333·55-s + 1.92·59-s − 1.59·61-s − 0.134·65-s + 1.39·67-s + 0.360·71-s + 0.812·73-s + 0.322·77-s − 0.171·79-s − 0.524·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 0.874T + 5T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 + 0.763T + 43T^{2} \) |
| 47 | \( 1 - 5.45T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 3.03T + 71T^{2} \) |
| 73 | \( 1 - 6.94T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 + 4.78T + 83T^{2} \) |
| 89 | \( 1 + 8.48T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−8.034049779544003165818549812882, −7.18516421446052085862803312594, −6.60312597456993023799686477112, −5.55540532522430471845560115726, −5.15763608087347194700895880092, −4.03233584795829395532867708854, −3.35196890872376221500329790374, −2.54446694756704912200377162084, −1.29759661102278597750703260628, 0,
1.29759661102278597750703260628, 2.54446694756704912200377162084, 3.35196890872376221500329790374, 4.03233584795829395532867708854, 5.15763608087347194700895880092, 5.55540532522430471845560115726, 6.60312597456993023799686477112, 7.18516421446052085862803312594, 8.034049779544003165818549812882
