| L(s) = 1 | − 1.55·2-s + 3-s + 0.404·4-s − 0.920·5-s − 1.55·6-s − 2.53·7-s + 2.47·8-s + 9-s + 1.42·10-s − 0.294·11-s + 0.404·12-s + 5.74·13-s + 3.93·14-s − 0.920·15-s − 4.64·16-s − 1.70·17-s − 1.55·18-s − 4.65·19-s − 0.372·20-s − 2.53·21-s + 0.456·22-s − 23-s + 2.47·24-s − 4.15·25-s − 8.90·26-s + 27-s − 1.02·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.577·3-s + 0.202·4-s − 0.411·5-s − 0.633·6-s − 0.959·7-s + 0.874·8-s + 0.333·9-s + 0.451·10-s − 0.0888·11-s + 0.116·12-s + 1.59·13-s + 1.05·14-s − 0.237·15-s − 1.16·16-s − 0.414·17-s − 0.365·18-s − 1.06·19-s − 0.0832·20-s − 0.553·21-s + 0.0974·22-s − 0.208·23-s + 0.504·24-s − 0.830·25-s − 1.74·26-s + 0.192·27-s − 0.194·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 5 | \( 1 + 0.920T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 + 0.294T + 11T^{2} \) |
| 13 | \( 1 - 5.74T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 31 | \( 1 - 4.26T + 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 1.68T + 47T^{2} \) |
| 53 | \( 1 - 3.05T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 + 7.06T + 67T^{2} \) |
| 71 | \( 1 - 0.554T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 9.84T + 83T^{2} \) |
| 89 | \( 1 + 0.967T + 89T^{2} \) |
| 97 | \( 1 - 3.92T + 97T^{2} \) |
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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
−8.753906590374105764576347176564, −8.278923464364643735833888399287, −7.45411220899958664583810697449, −6.63317240203580595558763509279, −5.84565631387789089212692150797, −4.30953130498294012319441422752, −3.82246299426895757573130600217, −2.62457899772300937635548164489, −1.37725616078221464571945029065, 0,
1.37725616078221464571945029065, 2.62457899772300937635548164489, 3.82246299426895757573130600217, 4.30953130498294012319441422752, 5.84565631387789089212692150797, 6.63317240203580595558763509279, 7.45411220899958664583810697449, 8.278923464364643735833888399287, 8.753906590374105764576347176564
