| L(s) = 1 | − 2.70·2-s + 5.29·4-s − 0.533·5-s + 0.354·7-s − 8.89·8-s + 1.44·10-s − 4.18·11-s − 3.72·13-s − 0.958·14-s + 13.4·16-s + 6.46·17-s − 1.31·19-s − 2.82·20-s + 11.3·22-s + 4.10·23-s − 4.71·25-s + 10.0·26-s + 1.87·28-s + 8.85·29-s + 5.11·31-s − 18.4·32-s − 17.4·34-s − 0.189·35-s + 5.41·37-s + 3.53·38-s + 4.74·40-s − 11.8·41-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.64·4-s − 0.238·5-s + 0.134·7-s − 3.14·8-s + 0.455·10-s − 1.26·11-s − 1.03·13-s − 0.256·14-s + 3.35·16-s + 1.56·17-s − 0.300·19-s − 0.631·20-s + 2.41·22-s + 0.856·23-s − 0.943·25-s + 1.97·26-s + 0.355·28-s + 1.64·29-s + 0.918·31-s − 3.26·32-s − 2.99·34-s − 0.0320·35-s + 0.890·37-s + 0.573·38-s + 0.750·40-s − 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 5 | \( 1 + 0.533T + 5T^{2} \) |
| 7 | \( 1 - 0.354T + 7T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 - 4.10T + 23T^{2} \) |
| 29 | \( 1 - 8.85T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 0.673T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 - 9.92T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 7.76T + 73T^{2} \) |
| 79 | \( 1 + 1.17T + 79T^{2} \) |
| 83 | \( 1 + 6.25T + 83T^{2} \) |
| 89 | \( 1 + 3.80T + 89T^{2} \) |
| 97 | \( 1 + 9.91T + 97T^{2} \) |
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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
−8.482156429975564215967680298331, −8.074515339812057871724672588293, −7.47286495479480768473868190380, −6.76816372179176117520332328635, −5.76285544868792190294152617574, −4.83648796948890486577216340126, −3.12266073347655452560347542325, −2.49736727634282596074893182307, −1.23462896609142952604409923683, 0,
1.23462896609142952604409923683, 2.49736727634282596074893182307, 3.12266073347655452560347542325, 4.83648796948890486577216340126, 5.76285544868792190294152617574, 6.76816372179176117520332328635, 7.47286495479480768473868190380, 8.074515339812057871724672588293, 8.482156429975564215967680298331
