| L(s) = 1 | − 2.89·3-s + 2.44·5-s − 0.245·7-s + 5.39·9-s + 4.68·13-s − 7.08·15-s − 2.54·17-s + 19-s + 0.710·21-s + 0.746·23-s + 0.974·25-s − 6.95·27-s + 2.58·29-s − 0.946·31-s − 0.599·35-s − 0.399·37-s − 13.5·39-s − 8.49·41-s − 8.39·43-s + 13.1·45-s + 4.63·47-s − 6.93·49-s + 7.37·51-s + 7.75·53-s − 2.89·57-s − 12.6·59-s − 4.11·61-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 1.09·5-s − 0.0926·7-s + 1.79·9-s + 1.30·13-s − 1.82·15-s − 0.617·17-s + 0.229·19-s + 0.155·21-s + 0.155·23-s + 0.194·25-s − 1.33·27-s + 0.479·29-s − 0.169·31-s − 0.101·35-s − 0.0657·37-s − 2.17·39-s − 1.32·41-s − 1.28·43-s + 1.96·45-s + 0.675·47-s − 0.991·49-s + 1.03·51-s + 1.06·53-s − 0.383·57-s − 1.65·59-s − 0.527·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 + 0.245T + 7T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 23 | \( 1 - 0.746T + 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 31 | \( 1 + 0.946T + 31T^{2} \) |
| 37 | \( 1 + 0.399T + 37T^{2} \) |
| 41 | \( 1 + 8.49T + 41T^{2} \) |
| 43 | \( 1 + 8.39T + 43T^{2} \) |
| 47 | \( 1 - 4.63T + 47T^{2} \) |
| 53 | \( 1 - 7.75T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 5.31T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 + 6.84T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 3.39T + 97T^{2} \) |
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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
−6.99081594312013002076834891672, −6.43212635399998609278924995209, −6.09144811420681977618901685395, −5.41251665682247239651924809900, −4.88083096746101716176701754967, −4.05488366932557120025135843832, −3.06465129931710360387606303166, −1.80909130161415372194786669653, −1.21875220586446939079448067967, 0,
1.21875220586446939079448067967, 1.80909130161415372194786669653, 3.06465129931710360387606303166, 4.05488366932557120025135843832, 4.88083096746101716176701754967, 5.41251665682247239651924809900, 6.09144811420681977618901685395, 6.43212635399998609278924995209, 6.99081594312013002076834891672
