L(s) = 1 | + (1.18 − 0.769i)2-s + (0.808 + 0.808i)3-s + (0.817 − 1.82i)4-s + (0.885 + 2.05i)5-s + (1.58 + 0.337i)6-s + (−0.771 + 0.771i)7-s + (−0.433 − 2.79i)8-s − 1.69i·9-s + (2.63 + 1.75i)10-s + 0.875i·11-s + (2.13 − 0.815i)12-s + (0.707 − 0.707i)13-s + (−0.322 + 1.50i)14-s + (−0.943 + 2.37i)15-s + (−2.66 − 2.98i)16-s + (4.91 + 4.91i)17-s + ⋯ |
L(s) = 1 | + (0.839 − 0.543i)2-s + (0.466 + 0.466i)3-s + (0.408 − 0.912i)4-s + (0.396 + 0.918i)5-s + (0.645 + 0.137i)6-s + (−0.291 + 0.291i)7-s + (−0.153 − 0.988i)8-s − 0.563i·9-s + (0.831 + 0.555i)10-s + 0.264i·11-s + (0.616 − 0.235i)12-s + (0.196 − 0.196i)13-s + (−0.0861 + 0.403i)14-s + (−0.243 + 0.613i)15-s + (−0.666 − 0.745i)16-s + (1.19 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.24734 - 0.277107i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.24734 - 0.277107i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.18 + 0.769i)T \) |
| 5 | \( 1 + (-0.885 - 2.05i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.808 - 0.808i)T + 3iT^{2} \) |
| 7 | \( 1 + (0.771 - 0.771i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.875iT - 11T^{2} \) |
| 17 | \( 1 + (-4.91 - 4.91i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.89T + 19T^{2} \) |
| 23 | \( 1 + (6.03 + 6.03i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.85iT - 29T^{2} \) |
| 31 | \( 1 - 5.45iT - 31T^{2} \) |
| 37 | \( 1 + (2.21 + 2.21i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.36T + 41T^{2} \) |
| 43 | \( 1 + (-4.34 - 4.34i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.81 - 3.81i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.18 - 2.18i)T - 53iT^{2} \) |
| 59 | \( 1 - 0.749T + 59T^{2} \) |
| 61 | \( 1 + 4.79T + 61T^{2} \) |
| 67 | \( 1 + (-10.0 + 10.0i)T - 67iT^{2} \) |
| 71 | \( 1 - 5.01iT - 71T^{2} \) |
| 73 | \( 1 + (-6.85 + 6.85i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.62T + 79T^{2} \) |
| 83 | \( 1 + (-2.66 - 2.66i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.68iT - 89T^{2} \) |
| 97 | \( 1 + (-6.65 - 6.65i)T + 97iT^{2} \) |
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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
−12.21749183032550536513395707636, −10.81474359291858662333049637464, −10.24350756475409623461790041293, −9.481718036920021204731180541685, −8.128202454691245312517302596348, −6.41786745411541645570402934453, −6.04658483699863496463866380833, −4.28444902502286175096684930488, −3.36970886045077268508952707211, −2.20747527047709293040728443710,
2.03130857408272735598983214035, 3.59826307756106261764956238888, 4.92423409291814848944065157349, 5.83712164118923681609555293492, 7.08121293514330968461404484841, 8.007936615773673317645027481353, 8.787372981915081224032946118891, 10.04760734136017112122979259608, 11.42113767022272828885965037390, 12.41262290844507645490067670553