| L(s) = 1 | − 2.15·2-s − 3-s + 2.66·4-s − 0.295·5-s + 2.15·6-s + 3.97·7-s − 1.42·8-s + 9-s + 0.637·10-s − 3.95·11-s − 2.66·12-s + 3.40·13-s − 8.58·14-s + 0.295·15-s − 2.23·16-s + 3.81·17-s − 2.15·18-s + 2.91·19-s − 0.785·20-s − 3.97·21-s + 8.53·22-s − 5.02·23-s + 1.42·24-s − 4.91·25-s − 7.35·26-s − 27-s + 10.5·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.577·3-s + 1.33·4-s − 0.132·5-s + 0.881·6-s + 1.50·7-s − 0.505·8-s + 0.333·9-s + 0.201·10-s − 1.19·11-s − 0.768·12-s + 0.945·13-s − 2.29·14-s + 0.0762·15-s − 0.559·16-s + 0.925·17-s − 0.508·18-s + 0.668·19-s − 0.175·20-s − 0.867·21-s + 1.81·22-s − 1.04·23-s + 0.291·24-s − 0.982·25-s − 1.44·26-s − 0.192·27-s + 1.99·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 5 | \( 1 + 0.295T + 5T^{2} \) |
| 7 | \( 1 - 3.97T + 7T^{2} \) |
| 11 | \( 1 + 3.95T + 11T^{2} \) |
| 13 | \( 1 - 3.40T + 13T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 23 | \( 1 + 5.02T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 - 3.10T + 37T^{2} \) |
| 41 | \( 1 + 5.07T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 + 9.95T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 + 9.75T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 6.56T + 83T^{2} \) |
| 89 | \( 1 + 9.87T + 89T^{2} \) |
| 97 | \( 1 - 3.28T + 97T^{2} \) |
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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
−7.66321787619244076893469747824, −7.34723888301529322801534989393, −6.00254129418246906129055560677, −5.66783041262664417845123620428, −4.73068539479441478657717873893, −4.02212144862559372843142035159, −2.73267751414130834588261327549, −1.72206739902716394625672434088, −1.16571712667026808338207981819, 0,
1.16571712667026808338207981819, 1.72206739902716394625672434088, 2.73267751414130834588261327549, 4.02212144862559372843142035159, 4.73068539479441478657717873893, 5.66783041262664417845123620428, 6.00254129418246906129055560677, 7.34723888301529322801534989393, 7.66321787619244076893469747824
