L(s) = 1 | + 2.89i·3-s + (−1.50 + 1.65i)5-s − 0.580i·7-s − 5.40·9-s − 0.0809·11-s + 3.02i·13-s + (−4.79 − 4.36i)15-s + 0.280i·17-s − 4.72·19-s + 1.68·21-s − i·23-s + (−0.470 − 4.97i)25-s − 6.98i·27-s − 1.38·29-s − 5.70·31-s + ⋯ |
L(s) = 1 | + 1.67i·3-s + (−0.673 + 0.739i)5-s − 0.219i·7-s − 1.80·9-s − 0.0243·11-s + 0.837i·13-s + (−1.23 − 1.12i)15-s + 0.0680i·17-s − 1.08·19-s + 0.367·21-s − 0.208i·23-s + (−0.0940 − 0.995i)25-s − 1.34i·27-s − 0.256·29-s − 1.02·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.263006 - 0.594912i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.263006 - 0.594912i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.50 - 1.65i)T \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 - 2.89iT - 3T^{2} \) |
| 7 | \( 1 + 0.580iT - 7T^{2} \) |
| 11 | \( 1 + 0.0809T + 11T^{2} \) |
| 13 | \( 1 - 3.02iT - 13T^{2} \) |
| 17 | \( 1 - 0.280iT - 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 - 2.61iT - 37T^{2} \) |
| 41 | \( 1 - 5.31T + 41T^{2} \) |
| 43 | \( 1 - 7.30iT - 43T^{2} \) |
| 47 | \( 1 + 12.2iT - 47T^{2} \) |
| 53 | \( 1 + 6.64iT - 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 5.99iT - 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 1.14iT - 73T^{2} \) |
| 79 | \( 1 - 1.68T + 79T^{2} \) |
| 83 | \( 1 - 9.86iT - 83T^{2} \) |
| 89 | \( 1 + 6.69T + 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
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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
−10.54834030967595575749563423885, −9.950928915229073112702744331861, −9.039492083810202413842914231624, −8.340789337133373128053362652484, −7.20009999622583674813040586676, −6.28130811978136145731494996247, −5.12633346697529769387180988266, −4.12784064604449215875548657385, −3.74329851962477145076990664930, −2.48810551028589288751792343147,
0.30376559036992195216469201814, 1.55841226145620979598485741212, 2.74498339185189820922124139413, 4.09608310547633295914565484978, 5.41914637849643281994933509527, 6.12323799682489271463934629646, 7.27274976111664464097785558415, 7.71315631758972242787420940457, 8.539257660097620878937470992715, 9.167213684582974392872057827820