L(s) = 1 | + 0.609·2-s + 3-s − 1.62·4-s + 3.04·5-s + 0.609·6-s + 7-s − 2.21·8-s + 9-s + 1.85·10-s − 2.44·11-s − 1.62·12-s − 3.77·13-s + 0.609·14-s + 3.04·15-s + 1.90·16-s + 3.61·17-s + 0.609·18-s + 7.07·19-s − 4.95·20-s + 21-s − 1.49·22-s − 6.33·23-s − 2.21·24-s + 4.26·25-s − 2.29·26-s + 27-s − 1.62·28-s + ⋯ |
L(s) = 1 | + 0.431·2-s + 0.577·3-s − 0.813·4-s + 1.36·5-s + 0.248·6-s + 0.377·7-s − 0.782·8-s + 0.333·9-s + 0.587·10-s − 0.737·11-s − 0.469·12-s − 1.04·13-s + 0.163·14-s + 0.785·15-s + 0.476·16-s + 0.875·17-s + 0.143·18-s + 1.62·19-s − 1.10·20-s + 0.218·21-s − 0.318·22-s − 1.32·23-s − 0.451·24-s + 0.853·25-s − 0.451·26-s + 0.192·27-s − 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 0.609T + 2T^{2} \) |
| 5 | \( 1 - 3.04T + 5T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 3.77T + 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + 6.33T + 23T^{2} \) |
| 29 | \( 1 + 9.12T + 29T^{2} \) |
| 31 | \( 1 + 9.05T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 9.22T + 47T^{2} \) |
| 53 | \( 1 - 8.85T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 5.97T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 - 3.75T + 97T^{2} \) |
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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
−7.54207681222791049681792338579, −6.85964960604459917395771560749, −5.63473038181931044551647678107, −5.35049376959258006770244054177, −5.00836899888955903150439512175, −3.68852580695820446295001076588, −3.29057043194399920745185712193, −2.16487335186250396653932805378, −1.60488006354703883681114719195, 0,
1.60488006354703883681114719195, 2.16487335186250396653932805378, 3.29057043194399920745185712193, 3.68852580695820446295001076588, 5.00836899888955903150439512175, 5.35049376959258006770244054177, 5.63473038181931044551647678107, 6.85964960604459917395771560749, 7.54207681222791049681792338579