| L(s) = 1 | + 1.97·3-s + 2.39·5-s + 0.915·9-s + 2.34·11-s + 1.41·13-s + 4.74·15-s + 2.31·17-s + 5.93·19-s − 6.90·23-s + 0.759·25-s − 4.12·27-s − 1.33·29-s − 4.98·31-s + 4.64·33-s + 9.14·37-s + 2.79·39-s + 41-s + 9.18·43-s + 2.19·45-s − 10.6·47-s + 4.58·51-s + 12.4·53-s + 5.63·55-s + 11.7·57-s − 5.58·59-s + 12.8·61-s + 3.38·65-s + ⋯ |
| L(s) = 1 | + 1.14·3-s + 1.07·5-s + 0.305·9-s + 0.708·11-s + 0.391·13-s + 1.22·15-s + 0.561·17-s + 1.36·19-s − 1.43·23-s + 0.151·25-s − 0.793·27-s − 0.247·29-s − 0.894·31-s + 0.809·33-s + 1.50·37-s + 0.447·39-s + 0.156·41-s + 1.40·43-s + 0.327·45-s − 1.55·47-s + 0.641·51-s + 1.71·53-s + 0.760·55-s + 1.55·57-s − 0.726·59-s + 1.64·61-s + 0.420·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.386794560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.386794560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 1.97T + 3T^{2} \) |
| 5 | \( 1 - 2.39T + 5T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 2.31T + 17T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + 1.33T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 43 | \( 1 - 9.18T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 5.58T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 16.3T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 3.76T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 - 7.17T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−7.84751307366683726311010489715, −7.36557824695187627930232766790, −6.31657771613562432391702545475, −5.83352544254070169506946592400, −5.17172928958582756409029029463, −3.93323411892291352947015080430, −3.55443829643869302446200640069, −2.54331215763391893952684646712, −1.96925874035511416696165409872, −1.03285076945488024077972682575,
1.03285076945488024077972682575, 1.96925874035511416696165409872, 2.54331215763391893952684646712, 3.55443829643869302446200640069, 3.93323411892291352947015080430, 5.17172928958582756409029029463, 5.83352544254070169506946592400, 6.31657771613562432391702545475, 7.36557824695187627930232766790, 7.84751307366683726311010489715
