| L(s) = 1 | + 2.82·3-s + 5-s − 7-s + 5.00·9-s − 4·11-s + 13-s + 2.82·15-s − 4.82·17-s + 1.17·19-s − 2.82·21-s + 1.17·23-s + 25-s + 5.65·27-s + 6·29-s + 9.65·31-s − 11.3·33-s − 35-s + 7.65·37-s + 2.82·39-s + 6.48·41-s + 11.3·43-s + 5.00·45-s − 11.3·47-s + 49-s − 13.6·51-s + 3.17·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 0.447·5-s − 0.377·7-s + 1.66·9-s − 1.20·11-s + 0.277·13-s + 0.730·15-s − 1.17·17-s + 0.268·19-s − 0.617·21-s + 0.244·23-s + 0.200·25-s + 1.08·27-s + 1.11·29-s + 1.73·31-s − 1.96·33-s − 0.169·35-s + 1.25·37-s + 0.452·39-s + 1.01·41-s + 1.72·43-s + 0.745·45-s − 1.65·47-s + 0.142·49-s − 1.91·51-s + 0.435·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.913728857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.913728857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 9.65T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 3.17T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.014024767934606547954354253573, −7.46359662234853616978496560907, −6.57655514032561892287641293779, −5.98592149011740776196442977598, −4.82840469214355588118969599284, −4.31079810338088371190190276284, −3.26669710906927377418460741795, −2.62678786718903676275713152351, −2.24797792380327376720337753192, −0.922505052848828195490066298557,
0.922505052848828195490066298557, 2.24797792380327376720337753192, 2.62678786718903676275713152351, 3.26669710906927377418460741795, 4.31079810338088371190190276284, 4.82840469214355588118969599284, 5.98592149011740776196442977598, 6.57655514032561892287641293779, 7.46359662234853616978496560907, 8.014024767934606547954354253573
