L(s) = 1 | + (1.15 + 0.820i)2-s + (0.655 + 1.88i)4-s + (2.47 − 1.42i)5-s + (2.58 + 0.555i)7-s + (−0.794 + 2.71i)8-s + (4.01 + 0.382i)10-s + (1.53 + 0.884i)11-s − 4.22i·13-s + (2.52 + 2.76i)14-s + (−3.14 + 2.47i)16-s + (−5.64 − 3.25i)17-s + (0.271 + 0.470i)19-s + (4.31 + 3.73i)20-s + (1.04 + 2.27i)22-s + (2.52 − 1.45i)23-s + ⋯ |
L(s) = 1 | + (0.814 + 0.579i)2-s + (0.327 + 0.944i)4-s + (1.10 − 0.637i)5-s + (0.977 + 0.210i)7-s + (−0.281 + 0.959i)8-s + (1.27 + 0.120i)10-s + (0.462 + 0.266i)11-s − 1.17i·13-s + (0.674 + 0.738i)14-s + (−0.785 + 0.618i)16-s + (−1.36 − 0.790i)17-s + (0.0623 + 0.107i)19-s + (0.964 + 0.835i)20-s + (0.221 + 0.485i)22-s + (0.525 − 0.303i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.86521 + 1.16504i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.86521 + 1.16504i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.15 - 0.820i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.58 - 0.555i)T \) |
good | 5 | \( 1 + (-2.47 + 1.42i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.53 - 0.884i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.22iT - 13T^{2} \) |
| 17 | \( 1 + (5.64 + 3.25i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.271 - 0.470i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.52 + 1.45i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.89T + 29T^{2} \) |
| 31 | \( 1 + (3.59 - 6.22i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.62 - 6.28i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 - 8.47iT - 43T^{2} \) |
| 47 | \( 1 + (0.518 + 0.897i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.96 - 5.14i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.373 + 0.646i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.14 - 1.81i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.4 + 7.78i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.60iT - 71T^{2} \) |
| 73 | \( 1 + (-7.52 - 4.34i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.876 + 0.506i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + (-6.49 + 3.74i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7.23iT - 97T^{2} \) |
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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
−10.61443643643821545349128171073, −9.284263759052925513055956935801, −8.741909810849843942342231823604, −7.76592310170244052936113394865, −6.82098619641296189779727377819, −5.79125116787559748598973141298, −5.12374984881988032392126670214, −4.43070229988572156826822796526, −2.87145474061339567131179054611, −1.70258476442651830635019780377,
1.67076531789894157140191283119, 2.31295447164421822555208193327, 3.83745150088761455141434926491, 4.65588176778982842815447563973, 5.80264274785470755628101068053, 6.45115206476432146659174904755, 7.33444896730450314959490649734, 8.874197611793910199401060066371, 9.512287927390035336401030577466, 10.50147363723614681478987627096