L(s) = 1 | + 0.941·2-s − 3-s − 1.11·4-s − 2.91·5-s − 0.941·6-s − 1.76·7-s − 2.93·8-s + 9-s − 2.74·10-s − 4.88·11-s + 1.11·12-s − 2.39·13-s − 1.65·14-s + 2.91·15-s − 0.531·16-s − 17-s + 0.941·18-s + 6.33·19-s + 3.24·20-s + 1.76·21-s − 4.60·22-s + 3.09·23-s + 2.93·24-s + 3.48·25-s − 2.25·26-s − 27-s + 1.96·28-s + ⋯ |
L(s) = 1 | + 0.665·2-s − 0.577·3-s − 0.556·4-s − 1.30·5-s − 0.384·6-s − 0.665·7-s − 1.03·8-s + 0.333·9-s − 0.867·10-s − 1.47·11-s + 0.321·12-s − 0.665·13-s − 0.443·14-s + 0.752·15-s − 0.132·16-s − 0.242·17-s + 0.221·18-s + 1.45·19-s + 0.725·20-s + 0.384·21-s − 0.981·22-s + 0.644·23-s + 0.598·24-s + 0.697·25-s − 0.442·26-s − 0.192·27-s + 0.370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 0.941T + 2T^{2} \) |
| 5 | \( 1 + 2.91T + 5T^{2} \) |
| 7 | \( 1 + 1.76T + 7T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 - 5.08T + 29T^{2} \) |
| 31 | \( 1 - 4.97T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 - 3.89T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 + 3.41T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 7.53T + 67T^{2} \) |
| 71 | \( 1 - 9.14T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 1.67T + 79T^{2} \) |
| 83 | \( 1 - 3.13T + 83T^{2} \) |
| 89 | \( 1 - 3.90T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.68205663666879387447344073483, −6.70201542027491638317350725521, −5.98094702338893072272052967143, −5.10190595576502318410598493736, −4.79820647449650463498017125493, −4.06068851472502943242515471374, −3.11159848442871096162019428054, −2.78239199288998033983606230110, −0.801706775614260952251323049293, 0,
0.801706775614260952251323049293, 2.78239199288998033983606230110, 3.11159848442871096162019428054, 4.06068851472502943242515471374, 4.79820647449650463498017125493, 5.10190595576502318410598493736, 5.98094702338893072272052967143, 6.70201542027491638317350725521, 7.68205663666879387447344073483