L(s) = 1 | + i·2-s + 1.73i·3-s + 4-s − 3.16·5-s − 1.73·6-s − 3.46·7-s + 3i·8-s − 2.99·9-s − 3.16i·10-s + (3.16 − i)11-s + 1.73i·12-s + (−1.73 − 3.16i)13-s − 3.46i·14-s − 5.47i·15-s − 16-s − 5.47·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.999i·3-s + 0.5·4-s − 1.41·5-s − 0.707·6-s − 1.30·7-s + 1.06i·8-s − 0.999·9-s − 1.00i·10-s + (0.953 − 0.301i)11-s + 0.499i·12-s + (−0.480 − 0.877i)13-s − 0.925i·14-s − 1.41i·15-s − 0.250·16-s − 1.32·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.198571 - 0.464882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.198571 - 0.464882i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 11 | \( 1 + (-3.16 + i)T \) |
| 13 | \( 1 + (1.73 + 3.16i)T \) |
good | 2 | \( 1 - iT - 2T^{2} \) |
| 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 5.47T + 29T^{2} \) |
| 31 | \( 1 - 5.47iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 - 9.48iT - 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 + 6.32T + 59T^{2} \) |
| 61 | \( 1 - 12.6iT - 61T^{2} \) |
| 67 | \( 1 + 5.47iT - 67T^{2} \) |
| 71 | \( 1 - 6.32T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 3.16iT - 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 3.16T + 89T^{2} \) |
| 97 | \( 1 - 10.9iT - 97T^{2} \) |
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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
−11.47864049980825241733616708460, −11.04771282814340186045236898685, −9.801923640471812292003460198840, −8.985751769620736374333826482723, −7.973469883231055382727689955399, −7.09304850622843174011324773436, −6.17857833194790267750639629755, −5.07554318920265461928165399321, −3.73418325082588303900545439535, −3.07818691102063799227753410754,
0.29784095610922973919854403482, 2.12957214607603629877565854009, 3.31793567612243685327676257964, 4.24838215781594132499593445922, 6.36321247415807729349639042350, 6.84608673949276279641429331879, 7.55612598194755510997238563189, 8.868391291802716535330493841668, 9.687102496887723034582800729663, 11.08874840003523902884433798766