| L(s) = 1 | − 1.87·3-s − 5-s + 4.15·7-s + 0.524·9-s − 2.27·13-s + 1.87·15-s − 2.80·17-s + 4.27·19-s − 7.80·21-s − 0.524·23-s + 25-s + 4.64·27-s + 3.75·29-s − 10.2·31-s − 4.15·35-s − 5.75·37-s + 4.27·39-s + 1.27·41-s + 6.96·43-s − 0.524·45-s − 10.4·47-s + 10.2·49-s + 5.26·51-s − 10.8·53-s − 8.03·57-s − 0.524·59-s + 8.47·61-s + ⋯ |
| L(s) = 1 | − 1.08·3-s − 0.447·5-s + 1.57·7-s + 0.174·9-s − 0.632·13-s + 0.484·15-s − 0.680·17-s + 0.981·19-s − 1.70·21-s − 0.109·23-s + 0.200·25-s + 0.894·27-s + 0.697·29-s − 1.84·31-s − 0.702·35-s − 0.946·37-s + 0.685·39-s + 0.199·41-s + 1.06·43-s − 0.0782·45-s − 1.52·47-s + 1.46·49-s + 0.737·51-s − 1.48·53-s − 1.06·57-s − 0.0683·59-s + 1.08·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 7 | \( 1 - 4.15T + 7T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 23 | \( 1 + 0.524T + 23T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 - 1.27T + 41T^{2} \) |
| 43 | \( 1 - 6.96T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 0.524T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 - 0.804T + 79T^{2} \) |
| 83 | \( 1 - 4.03T + 83T^{2} \) |
| 89 | \( 1 - 7.80T + 89T^{2} \) |
| 97 | \( 1 - 2.27T + 97T^{2} \) |
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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
−7.78426710909296396077531547271, −7.26656378001658168549144649870, −6.44310483888822543422705977202, −5.50115034486687796686463433972, −5.01294993360133391370697453226, −4.50887789449460731836108781181, −3.43547284125106784883258201782, −2.22207871065707618178622017280, −1.22955211254888184714218590206, 0,
1.22955211254888184714218590206, 2.22207871065707618178622017280, 3.43547284125106784883258201782, 4.50887789449460731836108781181, 5.01294993360133391370697453226, 5.50115034486687796686463433972, 6.44310483888822543422705977202, 7.26656378001658168549144649870, 7.78426710909296396077531547271
