L(s) = 1 | + (−0.495 + 0.285i)2-s + (−1.70 + 0.285i)3-s + (−0.836 + 1.44i)4-s + (0.764 − 0.630i)6-s + (−1.23 + 0.714i)7-s − 2.10i·8-s + (2.83 − 0.977i)9-s + (−1.33 − 2.31i)11-s + (1.01 − 2.71i)12-s + (−4.04 − 2.33i)13-s + (0.408 − 0.707i)14-s + (−1.07 − 1.85i)16-s − 2.67i·17-s + (−1.12 + 1.29i)18-s − 4.67·19-s + ⋯ |
L(s) = 1 | + (−0.350 + 0.202i)2-s + (−0.986 + 0.165i)3-s + (−0.418 + 0.724i)4-s + (0.312 − 0.257i)6-s + (−0.467 + 0.269i)7-s − 0.742i·8-s + (0.945 − 0.325i)9-s + (−0.402 − 0.697i)11-s + (0.292 − 0.783i)12-s + (−1.12 − 0.648i)13-s + (0.109 − 0.189i)14-s + (−0.267 − 0.464i)16-s − 0.648i·17-s + (−0.265 + 0.305i)18-s − 1.07·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0412628 - 0.0783057i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0412628 - 0.0783057i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.70 - 0.285i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.495 - 0.285i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1.23 - 0.714i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.33 + 2.31i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.04 + 2.33i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2.67iT - 17T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 + (-5.12 - 2.95i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.74 + 8.21i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.81iT - 37T^{2} \) |
| 41 | \( 1 + (-0.735 + 1.27i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.408 + 0.235i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.02 - 3.47i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 1.14iT - 53T^{2} \) |
| 59 | \( 1 + (0.571 - 0.990i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.26 - 2.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.70 + 3.29i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 1.71iT - 73T^{2} \) |
| 79 | \( 1 + (0.143 + 0.249i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.71 - 2.14i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + (6.78 - 3.91i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.97615453576407690212822351983, −10.95724754823836543799657851218, −9.889811303182592396223879625102, −9.096005978355202727248146158325, −7.81898512002321956233472310169, −6.92056610972392828954913845849, −5.67160354249808067054995191971, −4.57745458865628653567726890710, −3.10011795885534956160573264503, −0.089625988368504343775529566565,
1.91912018750320778659264059452, 4.37971992086850272606877256482, 5.26468119922338314541740162259, 6.46900502512227659582461474997, 7.38602051231870354288663445135, 8.941607289859662366273834159564, 9.916049306508257672284360568073, 10.56234938887180853812903070166, 11.40107957610659832454538645099, 12.68555663391719179951156126051