| L(s) = 1 | + 1.37·3-s + 1.97·5-s − 2.44·7-s − 1.09·9-s − 3.43·11-s + 4.69·13-s + 2.72·15-s + 1.90·17-s + 1.23·19-s − 3.36·21-s − 6.49·23-s − 1.10·25-s − 5.65·27-s − 5.54·29-s − 4.88·31-s − 4.73·33-s − 4.81·35-s − 10.2·37-s + 6.47·39-s − 5.60·41-s + 2.85·43-s − 2.16·45-s + 1.80·47-s − 1.04·49-s + 2.62·51-s + 2.04·53-s − 6.77·55-s + ⋯ |
| L(s) = 1 | + 0.796·3-s + 0.882·5-s − 0.922·7-s − 0.366·9-s − 1.03·11-s + 1.30·13-s + 0.702·15-s + 0.461·17-s + 0.282·19-s − 0.734·21-s − 1.35·23-s − 0.221·25-s − 1.08·27-s − 1.02·29-s − 0.876·31-s − 0.824·33-s − 0.814·35-s − 1.68·37-s + 1.03·39-s − 0.875·41-s + 0.434·43-s − 0.322·45-s + 0.264·47-s − 0.148·49-s + 0.367·51-s + 0.280·53-s − 0.914·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 - 1.37T + 3T^{2} \) |
| 5 | \( 1 - 1.97T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 5.54T + 29T^{2} \) |
| 31 | \( 1 + 4.88T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 - 1.80T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 - 1.02T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 9.64T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−8.177187871039468361191662247051, −7.38696311022926448605361549916, −6.47800471872512986131415016551, −5.65800348733173796378417175147, −5.44045495767852663358880163890, −3.79704051966393730680536035594, −3.42424418399327095164634090606, −2.44491105675455185585088522138, −1.71091608164255320629422460618, 0,
1.71091608164255320629422460618, 2.44491105675455185585088522138, 3.42424418399327095164634090606, 3.79704051966393730680536035594, 5.44045495767852663358880163890, 5.65800348733173796378417175147, 6.47800471872512986131415016551, 7.38696311022926448605361549916, 8.177187871039468361191662247051
