| L(s) = 1 | + 2.30·3-s + 3.30·5-s + 7-s + 2.30·9-s − 2.60·11-s + 4·13-s + 7.60·15-s + 17-s − 2.60·19-s + 2.30·21-s + 6·23-s + 5.90·25-s − 1.60·27-s + 1.39·29-s − 5.90·31-s − 6·33-s + 3.30·35-s − 2.60·37-s + 9.21·39-s + 4.30·41-s − 6.30·43-s + 7.60·45-s + 5.21·47-s + 49-s + 2.30·51-s + 11.3·53-s − 8.60·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 1.47·5-s + 0.377·7-s + 0.767·9-s − 0.785·11-s + 1.10·13-s + 1.96·15-s + 0.242·17-s − 0.597·19-s + 0.502·21-s + 1.25·23-s + 1.18·25-s − 0.308·27-s + 0.258·29-s − 1.06·31-s − 1.04·33-s + 0.558·35-s − 0.428·37-s + 1.47·39-s + 0.671·41-s − 0.961·43-s + 1.13·45-s + 0.760·47-s + 0.142·49-s + 0.322·51-s + 1.55·53-s − 1.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.015656162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.015656162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 - 4.30T + 41T^{2} \) |
| 43 | \( 1 + 6.30T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 - 8.30T + 61T^{2} \) |
| 67 | \( 1 - 4.51T + 67T^{2} \) |
| 71 | \( 1 - 6.60T + 71T^{2} \) |
| 73 | \( 1 - 4.90T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 - 2.78T + 89T^{2} \) |
| 97 | \( 1 - 0.697T + 97T^{2} \) |
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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
−8.145117245576285035611847798654, −7.20408426425167702053093679898, −6.59639545805124625488555263426, −5.55238750926577033356981973439, −5.34957690577874002576678040670, −4.13618802409150009926181298607, −3.36622449207444284263608943835, −2.52108343743143549413464637703, −2.04707138144833725594222833521, −1.12312830830933072074763878498,
1.12312830830933072074763878498, 2.04707138144833725594222833521, 2.52108343743143549413464637703, 3.36622449207444284263608943835, 4.13618802409150009926181298607, 5.34957690577874002576678040670, 5.55238750926577033356981973439, 6.59639545805124625488555263426, 7.20408426425167702053093679898, 8.145117245576285035611847798654
