| L(s) = 1 | + 0.585·5-s − 2·11-s − 5.41·13-s + 6.24·17-s + 2.82·19-s + 3.65·23-s − 4.65·25-s + 1.17·29-s − 6.82·31-s − 4·37-s − 2.24·41-s + 5.65·43-s − 2.82·47-s + 2·53-s − 1.17·55-s + 6.82·59-s − 3.75·61-s − 3.17·65-s − 5.65·67-s − 13.3·71-s + 5.89·73-s − 2.34·79-s + 15.3·83-s + 3.65·85-s − 5.75·89-s + 1.65·95-s − 5.41·97-s + ⋯ |
| L(s) = 1 | + 0.261·5-s − 0.603·11-s − 1.50·13-s + 1.51·17-s + 0.648·19-s + 0.762·23-s − 0.931·25-s + 0.217·29-s − 1.22·31-s − 0.657·37-s − 0.350·41-s + 0.862·43-s − 0.412·47-s + 0.274·53-s − 0.157·55-s + 0.888·59-s − 0.481·61-s − 0.393·65-s − 0.691·67-s − 1.58·71-s + 0.690·73-s − 0.263·79-s + 1.68·83-s + 0.396·85-s − 0.610·89-s + 0.169·95-s − 0.549·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 3.75T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 5.89T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 5.75T + 89T^{2} \) |
| 97 | \( 1 + 5.41T + 97T^{2} \) |
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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
−7.48236774658715995686815740232, −7.16956921683940196291798479440, −6.07532791639906062678841387694, −5.29467247491462988957264445627, −5.05405142391636705100854410244, −3.89478895266724071244657431064, −3.07798590356295717196796986122, −2.34913880795537966223492470966, −1.31309828122034037608229063502, 0,
1.31309828122034037608229063502, 2.34913880795537966223492470966, 3.07798590356295717196796986122, 3.89478895266724071244657431064, 5.05405142391636705100854410244, 5.29467247491462988957264445627, 6.07532791639906062678841387694, 7.16956921683940196291798479440, 7.48236774658715995686815740232
