| L(s) = 1 | − 5-s + 2.60·7-s + 4.60·11-s + 6.60·13-s − 1.60·17-s − 3.60·19-s − 3·23-s + 25-s − 1.39·29-s + 5.60·31-s − 2.60·35-s + 2·37-s + 4.60·41-s − 0.605·43-s − 9.21·47-s − 0.211·49-s − 1.60·53-s − 4.60·55-s − 1.39·59-s − 4.21·61-s − 6.60·65-s + 0.788·67-s + 7.39·71-s + 12.6·73-s + 12·77-s + 11.6·79-s + 3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.984·7-s + 1.38·11-s + 1.83·13-s − 0.389·17-s − 0.827·19-s − 0.625·23-s + 0.200·25-s − 0.258·29-s + 1.00·31-s − 0.440·35-s + 0.328·37-s + 0.719·41-s − 0.0923·43-s − 1.34·47-s − 0.0301·49-s − 0.220·53-s − 0.621·55-s − 0.181·59-s − 0.539·61-s − 0.819·65-s + 0.0963·67-s + 0.877·71-s + 1.47·73-s + 1.36·77-s + 1.30·79-s + 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.143064918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.143064918\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.60T + 41T^{2} \) |
| 43 | \( 1 + 0.605T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 + 1.39T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 - 0.788T + 67T^{2} \) |
| 71 | \( 1 - 7.39T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.860625114534061401680084711761, −8.356920610328689021621545649650, −7.73145558930634095324664582829, −6.43275139509929433468228465660, −6.27050832250318788228252721410, −4.90464937006232093368101117830, −4.12335232822114281437710012430, −3.52226796508639220118477455658, −1.98243830582799593825103542093, −1.05144717746452817314141476217,
1.05144717746452817314141476217, 1.98243830582799593825103542093, 3.52226796508639220118477455658, 4.12335232822114281437710012430, 4.90464937006232093368101117830, 6.27050832250318788228252721410, 6.43275139509929433468228465660, 7.73145558930634095324664582829, 8.356920610328689021621545649650, 8.860625114534061401680084711761
