L(s) = 1 | + (0.239 + 0.892i)2-s − 1.14i·3-s + (0.993 − 0.573i)4-s + (1.02 − 3.82i)5-s + (1.01 − 0.272i)6-s + (2.05 + 2.05i)8-s + 1.69·9-s + 3.65·10-s + (1.48 + 1.48i)11-s + (−0.654 − 1.13i)12-s + (−3.41 + 1.16i)13-s + (−4.36 − 1.16i)15-s + (−0.195 + 0.339i)16-s + (1.58 + 2.74i)17-s + (0.405 + 1.51i)18-s + (0.825 + 0.825i)19-s + ⋯ |
L(s) = 1 | + (0.169 + 0.630i)2-s − 0.658i·3-s + (0.496 − 0.286i)4-s + (0.458 − 1.71i)5-s + (0.415 − 0.111i)6-s + (0.726 + 0.726i)8-s + 0.565·9-s + 1.15·10-s + (0.448 + 0.448i)11-s + (−0.188 − 0.327i)12-s + (−0.946 + 0.324i)13-s + (−1.12 − 0.301i)15-s + (−0.0489 + 0.0847i)16-s + (0.384 + 0.666i)17-s + (0.0956 + 0.357i)18-s + (0.189 + 0.189i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.99634 - 0.851469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99634 - 0.851469i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.41 - 1.16i)T \) |
good | 2 | \( 1 + (-0.239 - 0.892i)T + (-1.73 + i)T^{2} \) |
| 3 | \( 1 + 1.14iT - 3T^{2} \) |
| 5 | \( 1 + (-1.02 + 3.82i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.48 - 1.48i)T + 11iT^{2} \) |
| 17 | \( 1 + (-1.58 - 2.74i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.825 - 0.825i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.26 + 1.88i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.584 + 1.01i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.88 + 1.30i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (4.26 - 1.14i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.85 - 6.93i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.91 + 1.10i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (11.2 + 3.00i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.236 - 0.0633i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 - 12.3iT - 61T^{2} \) |
| 67 | \( 1 + (-7.28 + 7.28i)T - 67iT^{2} \) |
| 71 | \( 1 + (-1.60 - 5.98i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-1.27 - 4.75i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.34 - 2.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.31 + 3.31i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.80 - 6.71i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-15.8 + 4.24i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.11049770485001476271623992962, −9.706383355109959987320145876799, −8.432511119017866203179141291992, −7.81252242637712676824624779409, −6.81775020781507705372716128017, −6.04622987839585996031257347965, −5.03862096705821865278074404266, −4.35001042620809931438720362011, −2.05393958108906544499421029877, −1.29549917381865342870264892946,
1.97497092963803680213792765841, 3.08550246903092148887468744185, 3.64952794186863107268846661354, 5.03546849809796121767544700596, 6.41134996568770310270819850996, 7.05128739379902292432363636250, 7.82141975739315154170628636283, 9.528668389072278017979630984938, 10.05695027322108141976496133891, 10.62348894471859454175736370599