L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00·6-s + (−1.22 − 1.22i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + 1.73i·14-s + 1.00·16-s + (−0.707 + 0.707i)18-s − i·19-s − 1.73·21-s + i·24-s + (−0.707 − 0.707i)27-s + 1.73i·29-s + (−0.707 + 0.707i)38-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00·6-s + (−1.22 − 1.22i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + 1.73i·14-s + 1.00·16-s + (−0.707 + 0.707i)18-s − i·19-s − 1.73·21-s + i·24-s + (−0.707 − 0.707i)27-s + 1.73i·29-s + (−0.707 + 0.707i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6430970722\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6430970722\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 7 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 1.73T + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 59 | \( 1 + 1.73iT - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.73iT - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.325567771850745694189639707708, −8.750192083732866016092214994003, −7.78994652114589879742154845747, −6.84654938424630517759177553754, −6.46694933084267962225753497477, −5.07173724360936179741679674889, −3.59223016141743187437532983317, −3.03056216960866377117179366632, −1.80024048932092364627599612678, −0.59167177883557388024564707434,
2.36667364711556143619609608140, 3.24271011178538409171159418243, 4.04195448403740388037631866470, 5.48943735445424056919466304459, 6.16530620142373789953552377171, 7.06750450824806579504866072291, 8.122811790030963499725550841337, 8.493220296441734375200923824700, 9.442087238422849462571634782075, 9.658208665004911117207361849298