L(s) = 1 | + (0.0341 + 0.0592i)2-s + (0.997 − 1.72i)4-s + (−1.33 + 2.30i)5-s + 0.273·8-s − 0.182·10-s + (−0.799 − 1.38i)11-s + (−2.62 + 4.54i)13-s + (−1.98 − 3.43i)16-s + 6.54·17-s + 1.90·19-s + (2.65 + 4.60i)20-s + (0.0546 − 0.0946i)22-s + (−1.53 + 2.65i)23-s + (−1.04 − 1.81i)25-s − 0.359·26-s + ⋯ |
L(s) = 1 | + (0.0241 + 0.0418i)2-s + (0.498 − 0.864i)4-s + (−0.595 + 1.03i)5-s + 0.0965·8-s − 0.0575·10-s + (−0.241 − 0.417i)11-s + (−0.728 + 1.26i)13-s + (−0.496 − 0.859i)16-s + 1.58·17-s + 0.436·19-s + (0.594 + 1.02i)20-s + (0.0116 − 0.0201i)22-s + (−0.319 + 0.554i)23-s + (−0.209 − 0.363i)25-s − 0.0704·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.514373112\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.514373112\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.0341 - 0.0592i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.33 - 2.30i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.799 + 1.38i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.62 - 4.54i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 + (1.53 - 2.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.19 - 5.53i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.35 - 5.81i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 + (-3.69 + 6.40i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.63 - 9.75i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.89 - 3.29i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 + (5.44 - 9.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.35 + 2.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.66 + 2.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 2.19T + 73T^{2} \) |
| 79 | \( 1 + (0.406 + 0.704i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.41 - 5.92i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 0.470T + 89T^{2} \) |
| 97 | \( 1 + (-2.57 - 4.46i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.788147260904815794243520530047, −9.215547344756735362520613442129, −7.79549792781424242046070636367, −7.30083117657027086381523525288, −6.54662242855960635800774106120, −5.68797804045781353474094472579, −4.78437373637815312233574581383, −3.49717788685068919880227049228, −2.66680863299218837315208653325, −1.32253872972867982353039561891,
0.66821507413414788206729704322, 2.35334188860510388446590338082, 3.35644355841654762211836124154, 4.31304775337178468415710300713, 5.18762447123844711124645292622, 6.13309197853799459556415182139, 7.54887684089448650471461127333, 7.76034798056222418669420784325, 8.401150619754803293212581721061, 9.530887351022964898348332559731