L(s) = 1 | − 1.80i·3-s + (1.73 − 1.41i)5-s − 2.12·7-s − 0.267·9-s + 4.11i·11-s − 2.72·13-s + (−2.55 − 3.13i)15-s − 1.79i·17-s + (−3.49 − 2.60i)19-s + 3.85i·21-s − 3.68·23-s + (0.999 − 4.89i)25-s − 4.93i·27-s − 10.5i·29-s − 4.42·31-s + ⋯ |
L(s) = 1 | − 1.04i·3-s + (0.774 − 0.632i)5-s − 0.804·7-s − 0.0893·9-s + 1.24i·11-s − 0.755·13-s + (−0.660 − 0.808i)15-s − 0.434i·17-s + (−0.801 − 0.598i)19-s + 0.840i·21-s − 0.769·23-s + (0.199 − 0.979i)25-s − 0.950i·27-s − 1.95i·29-s − 0.795·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9739563838\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9739563838\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.73 + 1.41i)T \) |
| 19 | \( 1 + (3.49 + 2.60i)T \) |
good | 3 | \( 1 + 1.80iT - 3T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 4.11iT - 11T^{2} \) |
| 13 | \( 1 + 2.72T + 13T^{2} \) |
| 17 | \( 1 + 1.79iT - 17T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 + 10.5iT - 29T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 3.85iT - 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 59 | \( 1 - 4.42T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 3.13iT - 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 13.3iT - 73T^{2} \) |
| 79 | \( 1 - 6.29T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 10.5iT - 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−9.142447298193076883518456689993, −8.218650474696105497206059157621, −7.23700171766470106853173579633, −6.79299247978995312776120619145, −5.94601297531961971564235921380, −4.97297763192495594811049919373, −4.05880252417567482816434871700, −2.38355720991727777871158675575, −1.92037403996185316258090263848, −0.34888951502564720774541769014,
1.87265260841611784151405627024, 3.25486938216496360998778786088, 3.63892959449769623288003084646, 4.96313997798982956636094836530, 5.74016717969238743839993687259, 6.50715013299919655127118890435, 7.29846496366295569649076566009, 8.618599001942703047621535261175, 9.135891926124347466954719730808, 10.12742547310071101964792052482