L(s) = 1 | + 1.41·5-s + i·13-s + 1.41i·17-s + 1.00·25-s + 1.41i·29-s − 1.41·41-s + 49-s − 1.41i·53-s + 2·61-s + 1.41i·65-s − 2i·73-s + 2.00i·85-s − 1.41·89-s − 2i·97-s + 1.41i·101-s + ⋯ |
L(s) = 1 | + 1.41·5-s + i·13-s + 1.41i·17-s + 1.00·25-s + 1.41i·29-s − 1.41·41-s + 49-s − 1.41i·53-s + 2·61-s + 1.41i·65-s − 2i·73-s + 2.00i·85-s − 1.41·89-s − 2i·97-s + 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.654725957\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.654725957\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + 2iT - T^{2} \) |
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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
−8.791945349429324236578656158242, −8.252967581979535579487127644827, −6.99886700193495919969474213049, −6.59194081367993176915922813907, −5.76568350278613785408235613965, −5.17925890340391871912981734212, −4.18917370526847337885111180948, −3.25729397615093100749994637434, −2.05292053106648034834927557275, −1.56461113898216032464234289170,
1.00109666865667709685295600867, 2.28565011217361134562623793462, 2.82898557991785583771130688416, 4.01747593245025883239363820870, 5.16302259819073229024949637286, 5.51664390314294887398807649123, 6.33634260604897680827011216815, 7.07265689867623155677646220288, 7.891696029393824017686814939047, 8.717298675291547585479845286562