| L(s) = 1 | + 1.07·2-s + 2.43·3-s − 0.848·4-s + 0.625·5-s + 2.60·6-s − 3.05·8-s + 2.91·9-s + 0.671·10-s + 0.708·11-s − 2.06·12-s + 1.52·15-s − 1.58·16-s + 3.34·17-s + 3.12·18-s − 5.20·19-s − 0.530·20-s + 0.760·22-s − 4.43·23-s − 7.43·24-s − 4.60·25-s − 0.214·27-s − 6.59·29-s + 1.63·30-s − 4.39·31-s + 4.41·32-s + 1.72·33-s + 3.58·34-s + ⋯ |
| L(s) = 1 | + 0.758·2-s + 1.40·3-s − 0.424·4-s + 0.279·5-s + 1.06·6-s − 1.08·8-s + 0.970·9-s + 0.212·10-s + 0.213·11-s − 0.595·12-s + 0.392·15-s − 0.395·16-s + 0.810·17-s + 0.736·18-s − 1.19·19-s − 0.118·20-s + 0.162·22-s − 0.924·23-s − 1.51·24-s − 0.921·25-s − 0.0413·27-s − 1.22·29-s + 0.297·30-s − 0.788·31-s + 0.780·32-s + 0.299·33-s + 0.615·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.07T + 2T^{2} \) |
| 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 - 0.625T + 5T^{2} \) |
| 11 | \( 1 - 0.708T + 11T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 + 6.59T + 29T^{2} \) |
| 31 | \( 1 + 4.39T + 31T^{2} \) |
| 37 | \( 1 + 0.423T + 37T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 + 0.697T + 53T^{2} \) |
| 59 | \( 1 - 9.86T + 59T^{2} \) |
| 61 | \( 1 + 4.69T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 5.08T + 73T^{2} \) |
| 79 | \( 1 + 3.91T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 0.202T + 97T^{2} \) |
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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
−7.64524372685035329056527111336, −6.73981153591155411150323252217, −5.88807933753342799839167610300, −5.38995884364385503500673821682, −4.37729534178311985678858709641, −3.75304530611601665059455828367, −3.36513141073420227044169073972, −2.34894022211159490656854624017, −1.71824944918448313662234595346, 0,
1.71824944918448313662234595346, 2.34894022211159490656854624017, 3.36513141073420227044169073972, 3.75304530611601665059455828367, 4.37729534178311985678858709641, 5.38995884364385503500673821682, 5.88807933753342799839167610300, 6.73981153591155411150323252217, 7.64524372685035329056527111336
