| L(s) = 1 | − 2.09·2-s + 2.05·3-s + 2.37·4-s − 3.21·5-s − 4.30·6-s + 7-s − 0.789·8-s + 1.23·9-s + 6.71·10-s − 1.05·11-s + 4.89·12-s − 6.03·13-s − 2.09·14-s − 6.60·15-s − 3.10·16-s − 0.303·17-s − 2.58·18-s − 7.88·19-s − 7.63·20-s + 2.05·21-s + 2.19·22-s − 5.99·23-s − 1.62·24-s + 5.30·25-s + 12.6·26-s − 3.63·27-s + 2.37·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.18·3-s + 1.18·4-s − 1.43·5-s − 1.75·6-s + 0.377·7-s − 0.279·8-s + 0.411·9-s + 2.12·10-s − 0.316·11-s + 1.41·12-s − 1.67·13-s − 0.559·14-s − 1.70·15-s − 0.775·16-s − 0.0735·17-s − 0.608·18-s − 1.80·19-s − 1.70·20-s + 0.449·21-s + 0.468·22-s − 1.25·23-s − 0.331·24-s + 1.06·25-s + 2.47·26-s − 0.699·27-s + 0.449·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2729306716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2729306716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 859 | \( 1 + T \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 + 0.303T + 17T^{2} \) |
| 19 | \( 1 + 7.88T + 19T^{2} \) |
| 23 | \( 1 + 5.99T + 23T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 - 0.931T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 + 1.55T + 47T^{2} \) |
| 53 | \( 1 - 0.950T + 53T^{2} \) |
| 59 | \( 1 - 8.70T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 + 6.34T + 67T^{2} \) |
| 71 | \( 1 - 0.621T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.966434718325074578624248168263, −7.73012032668919898939444875414, −7.35051768634464595186300665859, −6.35972722408977069680177007800, −4.99482000836807940021943925417, −4.21723754881495406339183206825, −3.56580933281388129657555390505, −2.30822094259305434192295982919, −2.06091252222670695306969377488, −0.30032590500871893705037075760,
0.30032590500871893705037075760, 2.06091252222670695306969377488, 2.30822094259305434192295982919, 3.56580933281388129657555390505, 4.21723754881495406339183206825, 4.99482000836807940021943925417, 6.35972722408977069680177007800, 7.35051768634464595186300665859, 7.73012032668919898939444875414, 7.966434718325074578624248168263
