| L(s) = 1 | − 3-s + 0.267·5-s − 0.732·7-s + 9-s − 4.73·11-s − 0.267·15-s − 2.26·17-s − 1.26·19-s + 0.732·21-s + 6.19·23-s − 4.92·25-s − 27-s + 2.46·29-s + 5.46·31-s + 4.73·33-s − 0.196·35-s + 10.4·37-s + 11.3·41-s − 7.66·43-s + 0.267·45-s + 8.19·47-s − 6.46·49-s + 2.26·51-s + 0.464·53-s − 1.26·55-s + 1.26·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.119·5-s − 0.276·7-s + 0.333·9-s − 1.42·11-s − 0.0691·15-s − 0.550·17-s − 0.290·19-s + 0.159·21-s + 1.29·23-s − 0.985·25-s − 0.192·27-s + 0.457·29-s + 0.981·31-s + 0.823·33-s − 0.0331·35-s + 1.72·37-s + 1.77·41-s − 1.16·43-s + 0.0399·45-s + 1.19·47-s − 0.923·49-s + 0.317·51-s + 0.0637·53-s − 0.170·55-s + 0.167·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.267T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 7.66T + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 - 0.464T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 + 9.73T + 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.54809138161434619546420290971, −6.65737059939517128006911156560, −6.11636260325591896090525449280, −5.39828670039819067948970479008, −4.73751766104934244951651133522, −4.07877547754605999136455124686, −2.88932420801284205849653402185, −2.39577903564991379108570500924, −1.08224449129108285091996838074, 0,
1.08224449129108285091996838074, 2.39577903564991379108570500924, 2.88932420801284205849653402185, 4.07877547754605999136455124686, 4.73751766104934244951651133522, 5.39828670039819067948970479008, 6.11636260325591896090525449280, 6.65737059939517128006911156560, 7.54809138161434619546420290971
