L(s) = 1 | − 3-s + 9-s − 5·11-s + 2·13-s − 2·17-s − 6·19-s + 23-s − 27-s − 3·29-s + 4·31-s + 5·33-s + 5·37-s − 2·39-s + 4·41-s + 7·43-s − 10·47-s + 2·51-s − 2·53-s + 6·57-s − 10·59-s − 8·61-s − 7·67-s − 69-s − 3·71-s + 2·73-s − 11·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.554·13-s − 0.485·17-s − 1.37·19-s + 0.208·23-s − 0.192·27-s − 0.557·29-s + 0.718·31-s + 0.870·33-s + 0.821·37-s − 0.320·39-s + 0.624·41-s + 1.06·43-s − 1.45·47-s + 0.280·51-s − 0.274·53-s + 0.794·57-s − 1.30·59-s − 1.02·61-s − 0.855·67-s − 0.120·69-s − 0.356·71-s + 0.234·73-s − 1.23·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2552017252\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2552017252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.00116280799067, −12.57226485450102, −11.99247745301655, −11.35045424410863, −11.04669476660478, −10.65504297484376, −10.26522874491860, −9.778987068207279, −9.063370719142402, −8.782501838051938, −8.079218762595088, −7.727796237816014, −7.328110330619326, −6.517345317963892, −6.190922394423248, −5.838595940976872, −5.143287805446282, −4.647086558505512, −4.321140199606438, −3.626503603968222, −2.850852291894635, −2.497562268304664, −1.772225151923347, −1.110639521386424, −0.1515080459357695,
0.1515080459357695, 1.110639521386424, 1.772225151923347, 2.497562268304664, 2.850852291894635, 3.626503603968222, 4.321140199606438, 4.647086558505512, 5.143287805446282, 5.838595940976872, 6.190922394423248, 6.517345317963892, 7.328110330619326, 7.727796237816014, 8.079218762595088, 8.782501838051938, 9.063370719142402, 9.778987068207279, 10.26522874491860, 10.65504297484376, 11.04669476660478, 11.35045424410863, 11.99247745301655, 12.57226485450102, 13.00116280799067