| L(s) = 1 | − 1.49·3-s − 3.21·5-s − 1.95·7-s − 0.755·9-s + 5.72·11-s − 6.02·13-s + 4.81·15-s − 17-s + 2.70·19-s + 2.92·21-s − 1.14·23-s + 5.32·25-s + 5.62·27-s − 2.95·29-s − 3.60·31-s − 8.57·33-s + 6.26·35-s + 1.95·37-s + 9.02·39-s − 5.74·41-s + 7.26·43-s + 2.42·45-s − 8.00·47-s − 3.19·49-s + 1.49·51-s + 13.8·53-s − 18.3·55-s + ⋯ |
| L(s) = 1 | − 0.864·3-s − 1.43·5-s − 0.737·7-s − 0.251·9-s + 1.72·11-s − 1.67·13-s + 1.24·15-s − 0.242·17-s + 0.620·19-s + 0.637·21-s − 0.239·23-s + 1.06·25-s + 1.08·27-s − 0.548·29-s − 0.647·31-s − 1.49·33-s + 1.05·35-s + 0.321·37-s + 1.44·39-s − 0.897·41-s + 1.10·43-s + 0.361·45-s − 1.16·47-s − 0.456·49-s + 0.209·51-s + 1.90·53-s − 2.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 + 6.02T + 13T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 + 2.95T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 - 7.26T + 43T^{2} \) |
| 47 | \( 1 + 8.00T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 + 0.256T + 83T^{2} \) |
| 89 | \( 1 - 5.77T + 89T^{2} \) |
| 97 | \( 1 - 4.67T + 97T^{2} \) |
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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
−7.38943800178146822722263429343, −6.74168179886972266159562225568, −6.30524466361464006858253703838, −5.28651333694159655605838188474, −4.72135342929751951560889946476, −3.81766176815685981396665812129, −3.41319824784280368781746140693, −2.26836698524869141937595489246, −0.818790509958938628049838663111, 0,
0.818790509958938628049838663111, 2.26836698524869141937595489246, 3.41319824784280368781746140693, 3.81766176815685981396665812129, 4.72135342929751951560889946476, 5.28651333694159655605838188474, 6.30524466361464006858253703838, 6.74168179886972266159562225568, 7.38943800178146822722263429343
