L(s) = 1 | + 0.874·2-s − 2.48·3-s − 1.23·4-s − 4.06·5-s − 2.17·6-s − 2.82·8-s + 3.16·9-s − 3.55·10-s + 5.27·11-s + 3.06·12-s + 3.57·13-s + 10.0·15-s − 0.00594·16-s + 4.81·17-s + 2.76·18-s − 5.76·19-s + 5.02·20-s + 4.61·22-s − 6.67·23-s + 7.02·24-s + 11.5·25-s + 3.12·26-s − 0.401·27-s + 2.57·29-s + 8.82·30-s − 7.55·31-s + 5.65·32-s + ⋯ |
L(s) = 1 | + 0.618·2-s − 1.43·3-s − 0.617·4-s − 1.81·5-s − 0.886·6-s − 1.00·8-s + 1.05·9-s − 1.12·10-s + 1.59·11-s + 0.884·12-s + 0.990·13-s + 2.60·15-s − 0.00148·16-s + 1.16·17-s + 0.651·18-s − 1.32·19-s + 1.12·20-s + 0.983·22-s − 1.39·23-s + 1.43·24-s + 2.30·25-s + 0.612·26-s − 0.0771·27-s + 0.478·29-s + 1.61·30-s − 1.35·31-s + 0.999·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 0.874T + 2T^{2} \) |
| 3 | \( 1 + 2.48T + 3T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 - 2.57T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 - 6.08T + 37T^{2} \) |
| 43 | \( 1 + 0.758T + 43T^{2} \) |
| 47 | \( 1 - 4.96T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 8.86T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 4.34T + 67T^{2} \) |
| 71 | \( 1 - 0.0215T + 71T^{2} \) |
| 73 | \( 1 - 4.03T + 73T^{2} \) |
| 79 | \( 1 + 1.59T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 + 1.60T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.585669022541447581237971194990, −8.054442843358674163176642828322, −6.91359433334724438418551177179, −6.22229450244393657915245690712, −5.57573085351837107896750749309, −4.42991709934159163684616395953, −4.06345857606614758911116639349, −3.44458035158344830351688838848, −1.08137463809202876393989093276, 0,
1.08137463809202876393989093276, 3.44458035158344830351688838848, 4.06345857606614758911116639349, 4.42991709934159163684616395953, 5.57573085351837107896750749309, 6.22229450244393657915245690712, 6.91359433334724438418551177179, 8.054442843358674163176642828322, 8.585669022541447581237971194990