| L(s) = 1 | − 2·7-s − 2·11-s − 4·13-s − 2·17-s + 4·19-s + 8·23-s + 10·29-s + 4·31-s + 8·43-s + 8·47-s − 3·49-s + 6·53-s + 14·59-s − 14·61-s + 4·67-s − 12·71-s − 6·73-s + 4·77-s − 12·79-s + 4·83-s + 12·89-s + 8·91-s + 14·97-s − 6·101-s + 14·103-s − 12·107-s + 2·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.603·11-s − 1.10·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1.85·29-s + 0.718·31-s + 1.21·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s + 1.82·59-s − 1.79·61-s + 0.488·67-s − 1.42·71-s − 0.702·73-s + 0.455·77-s − 1.35·79-s + 0.439·83-s + 1.27·89-s + 0.838·91-s + 1.42·97-s − 0.597·101-s + 1.37·103-s − 1.16·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403338159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403338159\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.270407886951524481778316794857, −8.615181679584005774951790009650, −7.51380370959353581607795133194, −7.02360313048233502050208617514, −6.10428552516702050498345256322, −5.13249432243264413574946933168, −4.45417944745640591028127050063, −3.08658133002906810213405205063, −2.55273956376703116430496016927, −0.797276142181345148530396500843,
0.797276142181345148530396500843, 2.55273956376703116430496016927, 3.08658133002906810213405205063, 4.45417944745640591028127050063, 5.13249432243264413574946933168, 6.10428552516702050498345256322, 7.02360313048233502050208617514, 7.51380370959353581607795133194, 8.615181679584005774951790009650, 9.270407886951524481778316794857
