L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s − 3·9-s + 10-s + 11-s + 6·13-s − 16-s − 6·17-s + 3·18-s + 4·19-s + 20-s − 22-s − 8·23-s + 25-s − 6·26-s − 10·29-s + 4·31-s − 5·32-s + 6·34-s + 3·36-s + 6·37-s − 4·38-s − 3·40-s + 10·41-s + 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s − 9-s + 0.316·10-s + 0.301·11-s + 1.66·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.917·19-s + 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s − 1.17·26-s − 1.85·29-s + 0.718·31-s − 0.883·32-s + 1.02·34-s + 1/2·36-s + 0.986·37-s − 0.648·38-s − 0.474·40-s + 1.56·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.472331941885432765313453840071, −8.014549326893853956920710154873, −7.15953560226550428107527456219, −6.06938362406582818529049175858, −5.54487290485120562077957854154, −4.13715512280589368797699521783, −3.92722234583031258102323612556, −2.51910169739503906426540213901, −1.21450635323236466034392715955, 0,
1.21450635323236466034392715955, 2.51910169739503906426540213901, 3.92722234583031258102323612556, 4.13715512280589368797699521783, 5.54487290485120562077957854154, 6.06938362406582818529049175858, 7.15953560226550428107527456219, 8.014549326893853956920710154873, 8.472331941885432765313453840071