L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 1.73·5-s + (0.499 + 0.866i)6-s + (0.633 + 1.09i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.49i)10-s + (0.633 − 1.09i)11-s − 0.999·12-s − 1.26·14-s + (−0.866 + 1.49i)15-s + (−0.5 + 0.866i)16-s + (−2.59 − 4.5i)17-s + 0.999·18-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.774·5-s + (0.204 + 0.353i)6-s + (0.239 + 0.415i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.273 − 0.474i)10-s + (0.191 − 0.331i)11-s − 0.288·12-s − 0.338·14-s + (−0.223 + 0.387i)15-s + (−0.125 + 0.216i)16-s + (−0.630 − 1.09i)17-s + 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9338133646\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9338133646\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + (-0.633 - 1.09i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.633 + 1.09i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.59 + 4.5i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.36 - 4.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.09 + 7.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.23 + 5.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.09 - 3.63i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (6.92 + 12i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.59 + 13.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.63 + 6.29i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.09 + 1.90i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 + (-4.73 + 8.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.379491776947813735105116489274, −8.895688429937901495646026016474, −7.987276873358146093495862311050, −7.44840132380951396286984624919, −6.58413143715596786274542641419, −5.64515965657425916307585467019, −4.61337771539561811427005324939, −3.47388006322466673863389024397, −2.14713353501892156329977930257, −0.49164293832461253000007305805,
1.41446469989974304690030244998, 2.87722101140396126600806197990, 3.87946565583440586531951309197, 4.47274577948414626852604407914, 5.65941499594437548263762881360, 7.22966169474984014512374531515, 7.59189150412754879257412102933, 8.770888853072020647075179534926, 9.190948255934492824595851426841, 10.19449624712978457186068286472