L(s) = 1 | − 1.41i·3-s + (−1.22 − 1.87i)5-s + 2.64·7-s + 0.999·9-s − 3.46i·11-s − 2.44·13-s + (−2.64 + 1.73i)15-s − 4.89·17-s + 6.48·19-s − 3.74i·21-s + (−2 + 4.58i)25-s − 5.65i·27-s − 6·29-s − 4.89·33-s + (−3.24 − 4.94i)35-s + ⋯ |
L(s) = 1 | − 0.816i·3-s + (−0.547 − 0.836i)5-s + 0.999·7-s + 0.333·9-s − 1.04i·11-s − 0.679·13-s + (−0.683 + 0.447i)15-s − 1.18·17-s + 1.48·19-s − 0.816i·21-s + (−0.400 + 0.916i)25-s − 1.08i·27-s − 1.11·29-s − 0.852·33-s + (−0.547 − 0.836i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.709829 - 1.15356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.709829 - 1.15356i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.22 + 1.87i)T \) |
| 7 | \( 1 - 2.64T \) |
good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 6.48T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.16iT - 37T^{2} \) |
| 41 | \( 1 + 7.48iT - 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 9.16iT - 53T^{2} \) |
| 59 | \( 1 - 6.48T + 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 - 5.29T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 - 7.48iT - 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−10.71804429083272845999522132439, −9.330119604821830656489217500813, −8.625350287036401069392625783099, −7.65066140703252779222772647020, −7.24251432204283317227784542936, −5.77241096183556337455047071193, −4.88999835374130574791403665863, −3.82272708189864610780889570173, −2.08888375537595174430978255752, −0.813000375327978010493087769450,
2.00814864148212019463150509512, 3.45154144694842310144784528733, 4.53623690152769229613701565204, 5.09686198619392933847890135670, 6.79152169651812233688237287224, 7.40829655235702933193987724011, 8.323884179205492512017177352780, 9.674346295921948672666287459928, 10.00425753946506646930910951676, 11.20065776633256985746365224858