| L(s) = 1 | + (−0.449 − 1.67i)3-s + (0.731 + 0.195i)5-s + (−2.52 + 0.800i)7-s + (−0.0183 + 0.0105i)9-s + (−1.18 − 4.42i)11-s + (−2.89 − 2.89i)13-s − 1.31i·15-s + (−2.28 − 1.31i)17-s + (5.38 + 1.44i)19-s + (2.47 + 3.87i)21-s + (−1.01 − 1.75i)23-s + (−3.83 − 2.21i)25-s + (−3.66 − 3.66i)27-s + (−0.209 + 0.209i)29-s + (−3.33 + 5.76i)31-s + ⋯ |
| L(s) = 1 | + (−0.259 − 0.969i)3-s + (0.326 + 0.0875i)5-s + (−0.953 + 0.302i)7-s + (−0.00611 + 0.00353i)9-s + (−0.357 − 1.33i)11-s + (−0.803 − 0.803i)13-s − 0.339i·15-s + (−0.554 − 0.319i)17-s + (1.23 + 0.330i)19-s + (0.540 + 0.845i)21-s + (−0.210 − 0.365i)23-s + (−0.766 − 0.442i)25-s + (−0.704 − 0.704i)27-s + (−0.0389 + 0.0389i)29-s + (−0.598 + 1.03i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.324938 - 0.840409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.324938 - 0.840409i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.52 - 0.800i)T \) |
| good | 3 | \( 1 + (0.449 + 1.67i)T + (-2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.731 - 0.195i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (1.18 + 4.42i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (2.89 + 2.89i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.28 + 1.31i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.38 - 1.44i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (1.01 + 1.75i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.209 - 0.209i)T - 29iT^{2} \) |
| 31 | \( 1 + (3.33 - 5.76i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.02 + 3.82i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 5.04T + 41T^{2} \) |
| 43 | \( 1 + (-3.79 + 3.79i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.53 - 4.39i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.7 + 2.87i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-5.23 + 1.40i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (1.56 - 5.84i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (9.24 - 2.47i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.25T + 71T^{2} \) |
| 73 | \( 1 + (-3.29 + 5.70i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.0 + 7.54i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.00 + 8.00i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.92 - 6.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.79iT - 97T^{2} \) |
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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
−10.72523126527730998766546482470, −9.892009473725889325882476505520, −8.954192943966840195411419561774, −7.82456927016990231861241323962, −7.00534931474256764169459113432, −6.05267152899354955142828586549, −5.38955000190444312289921877399, −3.51143488428732201870873244840, −2.40384276136671268985026127546, −0.55855915757837480912561132364,
2.16582940011033050433589062024, 3.76024470794458480840564081428, 4.64475152855554054125402119202, 5.57148395986465933358317318301, 6.89316504724996210019882323842, 7.56833244605787819581721819719, 9.323002553141412789289061066399, 9.660730196591650028948706829895, 10.24252173969870576347373366621, 11.33673934796822222414495007935
