L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.17 − 2.02i)3-s + (−0.499 + 0.866i)4-s + (1.17 − 2.02i)6-s − 1.34·7-s − 0.999·8-s + (−1.23 + 2.14i)9-s + 3.25·11-s + 2.34·12-s + (−0.745 + 1.29i)13-s + (−0.670 − 1.16i)14-s + (−0.5 − 0.866i)16-s + (−3.29 − 5.70i)17-s − 2.47·18-s + (−1.25 − 4.17i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.675 − 1.17i)3-s + (−0.249 + 0.433i)4-s + (0.477 − 0.827i)6-s − 0.506·7-s − 0.353·8-s + (−0.413 + 0.715i)9-s + 0.982·11-s + 0.675·12-s + (−0.206 + 0.358i)13-s + (−0.179 − 0.310i)14-s + (−0.125 − 0.216i)16-s + (−0.798 − 1.38i)17-s − 0.584·18-s + (−0.288 − 0.957i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.156468 - 0.502644i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.156468 - 0.502644i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (1.25 + 4.17i)T \) |
good | 3 | \( 1 + (1.17 + 2.02i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + (0.745 - 1.29i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.29 + 5.70i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (1.07 - 1.86i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.65 - 4.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 + (2.79 + 4.83i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.81 + 6.61i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.465 + 0.806i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.54 - 7.87i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.0837 - 0.145i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.84 - 10.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.53 + 11.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.59 + 6.22i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.62 + 13.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.75 - 13.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.313T + 83T^{2} \) |
| 89 | \( 1 + (-4.90 + 8.49i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.01 - 10.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.314218854422632137736086335610, −8.955170105439866059913354616507, −7.55060992653509944457734587166, −6.90102932533418241182685774559, −6.57387104156464530558290149617, −5.54985079889287640726008489041, −4.61030327890411678957072796908, −3.34131071022528487651731579187, −1.87812978776303437628496931079, −0.22941756732342328381337363808,
1.82650470358016749049197294283, 3.48382618567641783788086731885, 4.05233677148491863526324005773, 4.91922342289937139727142559455, 5.98611176967760087324284798941, 6.50216435518491261012037012450, 8.092083896779400557701038175739, 9.059504878857451510971891097398, 9.918416802664559215861266758101, 10.28842342571434702047272842513