| L(s) = 1 | − 2-s + 0.581·3-s + 4-s − 2.21·5-s − 0.581·6-s + 1.23·7-s − 8-s − 2.66·9-s + 2.21·10-s − 2.13·11-s + 0.581·12-s − 1.70·13-s − 1.23·14-s − 1.28·15-s + 16-s + 3.71·17-s + 2.66·18-s + 1.68·19-s − 2.21·20-s + 0.720·21-s + 2.13·22-s + 23-s − 0.581·24-s − 0.112·25-s + 1.70·26-s − 3.29·27-s + 1.23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.335·3-s + 0.5·4-s − 0.988·5-s − 0.237·6-s + 0.468·7-s − 0.353·8-s − 0.887·9-s + 0.699·10-s − 0.644·11-s + 0.167·12-s − 0.472·13-s − 0.331·14-s − 0.332·15-s + 0.250·16-s + 0.901·17-s + 0.627·18-s + 0.386·19-s − 0.494·20-s + 0.157·21-s + 0.455·22-s + 0.208·23-s − 0.118·24-s − 0.0224·25-s + 0.334·26-s − 0.633·27-s + 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8230647654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8230647654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 0.581T + 3T^{2} \) |
| 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 2.13T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 - 1.68T + 19T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 1.60T + 37T^{2} \) |
| 41 | \( 1 + 7.44T + 41T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 - 9.01T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 0.596T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 - 3.45T + 71T^{2} \) |
| 73 | \( 1 + 2.66T + 73T^{2} \) |
| 79 | \( 1 + 8.04T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 - 2.02T + 97T^{2} \) |
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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
−8.040009601626300592488937757717, −7.65737937527180389798202774660, −7.02350072492928624264567231625, −5.90544083528726085900917713184, −5.30050592674885088019975126783, −4.39606355316986500324688496438, −3.37889503971488580804365738501, −2.83861944523948867839957477306, −1.79765657836134855814300601244, −0.50716451610930523969423816181,
0.50716451610930523969423816181, 1.79765657836134855814300601244, 2.83861944523948867839957477306, 3.37889503971488580804365738501, 4.39606355316986500324688496438, 5.30050592674885088019975126783, 5.90544083528726085900917713184, 7.02350072492928624264567231625, 7.65737937527180389798202774660, 8.040009601626300592488937757717
