L(s) = 1 | − 2-s − i·3-s + 4-s + 3.51i·5-s + i·6-s + i·7-s − 8-s − 9-s − 3.51i·10-s − 6.05i·11-s − i·12-s + 3.78·13-s − i·14-s + 3.51·15-s + 16-s + (−2.37 + 3.37i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.5·4-s + 1.57i·5-s + 0.408i·6-s + 0.377i·7-s − 0.353·8-s − 0.333·9-s − 1.11i·10-s − 1.82i·11-s − 0.288i·12-s + 1.04·13-s − 0.267i·14-s + 0.906·15-s + 0.250·16-s + (−0.575 + 0.817i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.928893 + 0.482264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.928893 + 0.482264i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| 17 | \( 1 + (2.37 - 3.37i)T \) |
good | 5 | \( 1 - 3.51iT - 5T^{2} \) |
| 11 | \( 1 + 6.05iT - 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 6.74iT - 23T^{2} \) |
| 29 | \( 1 + 1.72iT - 29T^{2} \) |
| 31 | \( 1 - 7.02iT - 31T^{2} \) |
| 37 | \( 1 - 5.78iT - 37T^{2} \) |
| 41 | \( 1 - 11.8iT - 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 7.02T + 47T^{2} \) |
| 53 | \( 1 + 2.76T + 53T^{2} \) |
| 59 | \( 1 + 1.45T + 59T^{2} \) |
| 61 | \( 1 + 7.84iT - 61T^{2} \) |
| 67 | \( 1 - 3.23T + 67T^{2} \) |
| 71 | \( 1 - 3.18iT - 71T^{2} \) |
| 73 | \( 1 - 11.0iT - 73T^{2} \) |
| 79 | \( 1 + 7.78iT - 79T^{2} \) |
| 83 | \( 1 + 6.33T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 + 18.0iT - 97T^{2} \) |
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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
−10.75126883833175983812325685005, −9.719320296751652973335392709488, −8.649070567453153200504497465575, −8.038291324814567552032412053330, −7.06728134249200936902659287351, −6.22872703441885118433889859408, −5.75249387414389591170828984997, −3.43078685096823278648402114678, −2.95496570997215110575952277786, −1.37874990499634031071927719431,
0.76714636267887904046489409571, 2.18420836734017886982857441711, 4.01640554798097323201334911059, 4.72078099750313206483712180818, 5.67456658837855992520538948259, 7.03516775642263473389889127105, 7.83034929497955841005646294486, 8.943921466943820806222120434787, 9.231901669618571514979491855522, 10.08815862346625588541611492765