| L(s) = 1 | − 0.331·2-s + 3-s − 1.89·4-s + 1.92·5-s − 0.331·6-s − 1.41·7-s + 1.28·8-s + 9-s − 0.638·10-s + 1.32·11-s − 1.89·12-s + 0.907·13-s + 0.468·14-s + 1.92·15-s + 3.35·16-s − 2.67·17-s − 0.331·18-s − 6.49·19-s − 3.64·20-s − 1.41·21-s − 0.439·22-s − 8.38·23-s + 1.28·24-s − 1.29·25-s − 0.300·26-s + 27-s + 2.67·28-s + ⋯ |
| L(s) = 1 | − 0.234·2-s + 0.577·3-s − 0.945·4-s + 0.861·5-s − 0.135·6-s − 0.534·7-s + 0.455·8-s + 0.333·9-s − 0.201·10-s + 0.400·11-s − 0.545·12-s + 0.251·13-s + 0.125·14-s + 0.497·15-s + 0.838·16-s − 0.648·17-s − 0.0781·18-s − 1.48·19-s − 0.814·20-s − 0.308·21-s − 0.0937·22-s − 1.74·23-s + 0.263·24-s − 0.258·25-s − 0.0589·26-s + 0.192·27-s + 0.505·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 0.331T + 2T^{2} \) |
| 5 | \( 1 - 1.92T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 - 0.907T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 + 8.38T + 23T^{2} \) |
| 29 | \( 1 - 0.500T + 29T^{2} \) |
| 37 | \( 1 - 1.49T + 37T^{2} \) |
| 41 | \( 1 + 6.27T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 3.53T + 53T^{2} \) |
| 59 | \( 1 + 6.88T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 - 6.50T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 8.33T + 97T^{2} \) |
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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
−8.414223973943552991414325881065, −8.027093467529584944698871702060, −6.74823748513468355736056854862, −6.20275214144224372723930431586, −5.31972712474650938774486873594, −4.21631711235866999174778906401, −3.78795947803005253307893495357, −2.45883688292584344587072892095, −1.60798122038188995558302368973, 0,
1.60798122038188995558302368973, 2.45883688292584344587072892095, 3.78795947803005253307893495357, 4.21631711235866999174778906401, 5.31972712474650938774486873594, 6.20275214144224372723930431586, 6.74823748513468355736056854862, 8.027093467529584944698871702060, 8.414223973943552991414325881065
