| L(s) = 1 | − 2.08·2-s + 2.88·3-s + 2.35·4-s − 4.18·5-s − 6.02·6-s − 1.99·7-s − 0.737·8-s + 5.33·9-s + 8.73·10-s + 3.06·11-s + 6.79·12-s + 0.158·13-s + 4.16·14-s − 12.0·15-s − 3.16·16-s − 3.54·17-s − 11.1·18-s + 4.28·19-s − 9.85·20-s − 5.76·21-s − 6.40·22-s − 1.05·23-s − 2.13·24-s + 12.5·25-s − 0.329·26-s + 6.74·27-s − 4.69·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.66·3-s + 1.17·4-s − 1.87·5-s − 2.45·6-s − 0.754·7-s − 0.260·8-s + 1.77·9-s + 2.76·10-s + 0.925·11-s + 1.96·12-s + 0.0438·13-s + 1.11·14-s − 3.12·15-s − 0.791·16-s − 0.859·17-s − 2.62·18-s + 0.984·19-s − 2.20·20-s − 1.25·21-s − 1.36·22-s − 0.219·23-s − 0.434·24-s + 2.50·25-s − 0.0646·26-s + 1.29·27-s − 0.888·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 3 | \( 1 - 2.88T + 3T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 7 | \( 1 + 1.99T + 7T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 - 0.158T + 13T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 - 4.28T + 19T^{2} \) |
| 23 | \( 1 + 1.05T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 41 | \( 1 + 0.110T + 41T^{2} \) |
| 43 | \( 1 - 3.87T + 43T^{2} \) |
| 47 | \( 1 + 3.80T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 3.45T + 67T^{2} \) |
| 71 | \( 1 - 8.33T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 1.18T + 79T^{2} \) |
| 83 | \( 1 - 0.476T + 83T^{2} \) |
| 89 | \( 1 - 1.85T + 89T^{2} \) |
| 97 | \( 1 + 5.50T + 97T^{2} \) |
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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
−8.161923695461186657050841202054, −7.62346623452777445706890013514, −7.18495520730890289614576227123, −6.49337388779571069295628713051, −4.66099154989360868274996440060, −3.81221980619194415084704495291, −3.42066983099922295829945897865, −2.41037639660213748439006982093, −1.24425294898757740945907223972, 0,
1.24425294898757740945907223972, 2.41037639660213748439006982093, 3.42066983099922295829945897865, 3.81221980619194415084704495291, 4.66099154989360868274996440060, 6.49337388779571069295628713051, 7.18495520730890289614576227123, 7.62346623452777445706890013514, 8.161923695461186657050841202054
