L(s) = 1 | − 1.41·3-s + (−0.707 − 0.707i)5-s + 1.00·9-s + (0.707 + 0.707i)13-s + (1.00 + 1.00i)15-s − 1.41i·17-s + (−0.707 − 0.707i)19-s + i·23-s − 29-s + (0.707 + 0.707i)31-s + (1 − i)37-s + (−1.00 − 1.00i)39-s − i·43-s + (−0.707 − 0.707i)45-s + (−0.707 + 0.707i)47-s + ⋯ |
L(s) = 1 | − 1.41·3-s + (−0.707 − 0.707i)5-s + 1.00·9-s + (0.707 + 0.707i)13-s + (1.00 + 1.00i)15-s − 1.41i·17-s + (−0.707 − 0.707i)19-s + i·23-s − 29-s + (0.707 + 0.707i)31-s + (1 − i)37-s + (−1.00 − 1.00i)39-s − i·43-s + (−0.707 − 0.707i)45-s + (−0.707 + 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2758604571\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2758604571\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + 1.41T + T^{2} \) |
| 5 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + (1 + i)T + iT^{2} \) |
| 73 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 97 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
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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
−9.001578997872869485161284936172, −7.83956529026855925783177674883, −7.18360106118855681313381775833, −6.31783587632359122136069814654, −5.67894842916647159486263792584, −4.66666187253310979703935214413, −4.42658649804905852969736366940, −3.10833031839903989285981777726, −1.50286681565648753878310539548, −0.23816130869538117204703114422,
1.42183979976451186754503062708, 2.97548874829959138728258373849, 3.95062780397026757742646800933, 4.65336878023744153409592434965, 5.85707555270533597432544401564, 6.16007631491381564935584718555, 6.86911477387296590596653272603, 8.003826388117821916502076740155, 8.279396441383810922036115835492, 9.642627470847927820011939432249