| L(s) = 1 | + 5-s − 4·11-s + 13-s − 6·17-s − 4·19-s − 8·23-s + 25-s + 6·29-s − 4·31-s − 2·37-s − 6·41-s + 4·43-s + 6·53-s − 4·55-s + 4·59-s − 10·61-s + 65-s + 8·67-s − 12·71-s − 6·73-s − 6·85-s − 6·89-s − 4·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s + 0.277·13-s − 1.45·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s + 0.609·43-s + 0.824·53-s − 0.539·55-s + 0.520·59-s − 1.28·61-s + 0.124·65-s + 0.977·67-s − 1.42·71-s − 0.702·73-s − 0.650·85-s − 0.635·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.39088528260870, −13.07228195408240, −12.46781195819797, −12.07834792424745, −11.55914024941530, −10.89297277428926, −10.55450138727682, −10.32340022353536, −9.702015665993668, −9.174461565166601, −8.599595539220631, −8.320583306666476, −7.836457654706924, −7.171864281117625, −6.662974412332968, −6.284023375826379, −5.659275941064801, −5.329907160902720, −4.551024940236076, −4.265309822650816, −3.640754072302668, −2.813910029294679, −2.393626551638999, −1.944806947268885, −1.247046522344355, 0, 0,
1.247046522344355, 1.944806947268885, 2.393626551638999, 2.813910029294679, 3.640754072302668, 4.265309822650816, 4.551024940236076, 5.329907160902720, 5.659275941064801, 6.284023375826379, 6.662974412332968, 7.171864281117625, 7.836457654706924, 8.320583306666476, 8.599595539220631, 9.174461565166601, 9.702015665993668, 10.32340022353536, 10.55450138727682, 10.89297277428926, 11.55914024941530, 12.07834792424745, 12.46781195819797, 13.07228195408240, 13.39088528260870
