L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.499 − 0.866i)6-s + (1.78 + 3.08i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (2.06 − 3.57i)11-s − 0.999·12-s + (3.34 + 1.35i)13-s + 3.56·14-s + (−0.5 + 0.866i)16-s + (2.56 + 4.43i)17-s − 0.999·18-s + (1.78 + 3.08i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.204 − 0.353i)6-s + (0.673 + 1.16i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.621 − 1.07i)11-s − 0.288·12-s + (0.926 + 0.375i)13-s + 0.951·14-s + (−0.125 + 0.216i)16-s + (0.621 + 1.07i)17-s − 0.235·18-s + (0.408 + 0.707i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.659701135\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.659701135\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.34 - 1.35i)T \) |
good | 7 | \( 1 + (-1.78 - 3.08i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.06 + 3.57i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.56 - 4.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.78 - 3.08i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.84 - 6.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.28 - 5.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 + (-2.06 + 3.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.12 + 3.67i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.28 - 3.95i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 7T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + (5.28 + 9.14i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.12 + 12.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.43 - 4.22i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 7.43T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 89 | \( 1 + (0.903 - 1.56i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.561 - 0.972i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.124112982967310450982164833589, −8.272401944892662011298026777447, −7.920983976234376704477325389338, −6.38084831624460273275557987701, −5.92005910430781055589221638894, −5.18842115578722503515741826327, −3.76915656329082789458868691119, −3.32331371544325126522387691653, −1.93544158199276962788318377983, −1.32785401268334758728197865330,
0.979687609296254591784940691191, 2.56809119830715925666651768577, 3.76474908758081342739552066432, 4.42834041583236543987221232228, 4.97401820693630656476477889974, 6.16811977613832819162866277202, 6.94892674098299596603020845599, 7.74990887223061392611039570598, 8.228794583378950439620842344270, 9.299798214976689710472462398622