| L(s) = 1 | + 2·5-s − 3·9-s + 11-s − 6·13-s + 2·17-s − 8·19-s − 25-s + 29-s − 4·31-s − 2·37-s + 2·41-s + 4·43-s − 6·45-s − 4·47-s − 10·53-s + 2·55-s + 12·59-s − 2·61-s − 12·65-s − 12·67-s − 8·71-s + 10·73-s + 9·81-s + 12·83-s + 4·85-s − 6·89-s − 16·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s + 0.301·11-s − 1.66·13-s + 0.485·17-s − 1.83·19-s − 1/5·25-s + 0.185·29-s − 0.718·31-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.894·45-s − 0.583·47-s − 1.37·53-s + 0.269·55-s + 1.56·59-s − 0.256·61-s − 1.48·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s + 81-s + 1.31·83-s + 0.433·85-s − 0.635·89-s − 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.06374259594754, −12.59103535476395, −12.20212357142192, −11.79778849687643, −11.15642787872634, −10.78101776060907, −10.23425043396062, −9.861009294249661, −9.340695629450310, −9.010880528816011, −8.429492293597096, −7.980378036130970, −7.416474904026661, −6.897382578842214, −6.348489119909208, −5.885286776685495, −5.571051323610310, −4.825237542634225, −4.586761206078808, −3.788189568353118, −3.171242913781247, −2.584876405936211, −2.067500758996990, −1.762650547212879, −0.6451970646980026, 0,
0.6451970646980026, 1.762650547212879, 2.067500758996990, 2.584876405936211, 3.171242913781247, 3.788189568353118, 4.586761206078808, 4.825237542634225, 5.571051323610310, 5.885286776685495, 6.348489119909208, 6.897382578842214, 7.416474904026661, 7.980378036130970, 8.429492293597096, 9.010880528816011, 9.340695629450310, 9.861009294249661, 10.23425043396062, 10.78101776060907, 11.15642787872634, 11.79778849687643, 12.20212357142192, 12.59103535476395, 13.06374259594754
