L(s) = 1 | + (−0.707 − 1.58i)3-s + 2.23i·5-s − 7-s + (−2.00 + 2.23i)9-s + 2.23i·11-s − 3.16i·13-s + (3.53 − 1.58i)15-s − 6.70i·17-s + (−3 − 3.16i)19-s + (0.707 + 1.58i)21-s − 4.47i·23-s + (4.94 + 1.58i)27-s + 5.65·29-s − 3.16i·31-s + (3.53 − 1.58i)33-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.912i)3-s + 0.999i·5-s − 0.377·7-s + (−0.666 + 0.745i)9-s + 0.674i·11-s − 0.877i·13-s + (0.912 − 0.408i)15-s − 1.62i·17-s + (−0.688 − 0.725i)19-s + (0.154 + 0.345i)21-s − 0.932i·23-s + (0.952 + 0.304i)27-s + 1.05·29-s − 0.567i·31-s + (0.615 − 0.275i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.482394 - 0.720778i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.482394 - 0.720778i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 1.58i)T \) |
| 19 | \( 1 + (3 + 3.16i)T \) |
good | 5 | \( 1 - 2.23iT - 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 3.16iT - 13T^{2} \) |
| 17 | \( 1 + 6.70iT - 17T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 + 3.16iT - 31T^{2} \) |
| 37 | \( 1 + 9.48iT - 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + 2.23iT - 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 6.32iT - 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 + 12.6iT - 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 3.16iT - 97T^{2} \) |
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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
−10.05128273705276275378165049658, −8.937099936863126864208714465678, −7.916194677846779579613007855675, −6.98132831895399362915872014492, −6.74268050916995412554314943872, −5.62277476298258352768183391930, −4.62411563386382750183457771526, −2.98842393226924914420396648086, −2.35457229626279657845647034775, −0.44714780020650638930014820179,
1.42286338387552004725334514200, 3.30853017385887234020676798615, 4.17062025127237771486620201882, 4.99759909603623267542343448495, 5.97421087106223824009225680366, 6.61383618787241119710815369609, 8.338857085958036522248221365306, 8.598598880898341357460979413473, 9.587837356339651592144247287874, 10.28075874266891010725600487646