L(s) = 1 | + (−0.920 − 1.59i)2-s + (−0.695 + 1.20i)4-s + (0.667 − 1.15i)5-s − 1.12·8-s − 2.45·10-s + (0.756 + 1.31i)11-s + (−2.58 + 4.48i)13-s + (2.42 + 4.19i)16-s + 1.54·17-s + 2.50·19-s + (0.927 + 1.60i)20-s + (1.39 − 2.41i)22-s + (−3.68 + 6.37i)23-s + (1.60 + 2.78i)25-s + 9.53·26-s + ⋯ |
L(s) = 1 | + (−0.650 − 1.12i)2-s + (−0.347 + 0.601i)4-s + (0.298 − 0.516i)5-s − 0.396·8-s − 0.777·10-s + (0.228 + 0.395i)11-s + (−0.717 + 1.24i)13-s + (0.605 + 1.04i)16-s + 0.375·17-s + 0.574·19-s + (0.207 + 0.359i)20-s + (0.296 − 0.514i)22-s + (−0.767 + 1.32i)23-s + (0.321 + 0.557i)25-s + 1.86·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9360186054\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9360186054\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.920 + 1.59i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.667 + 1.15i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.756 - 1.31i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.58 - 4.48i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.54T + 17T^{2} \) |
| 19 | \( 1 - 2.50T + 19T^{2} \) |
| 23 | \( 1 + (3.68 - 6.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0309 - 0.0536i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.92 + 3.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.563T + 37T^{2} \) |
| 41 | \( 1 + (4.51 - 7.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.09 - 8.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.75 + 8.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 1.51T + 53T^{2} \) |
| 59 | \( 1 + (4.22 - 7.31i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.61 + 2.80i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.46 - 6.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 2.75T + 73T^{2} \) |
| 79 | \( 1 + (-2.95 - 5.12i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.80 + 4.85i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 1.40T + 89T^{2} \) |
| 97 | \( 1 + (6.09 + 10.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.572670141317687827346762093100, −9.279521498837825969041309542807, −8.244412992290689241993417715265, −7.33078731178907457006090007189, −6.29341126302335551631423739058, −5.30416237954606587029225375068, −4.28390274191064982941528267331, −3.20437650858647280320172722692, −2.03842732221298560253535496416, −1.25758442179621031239041700480,
0.51905629182881622095124161830, 2.51792398163278523428888847970, 3.41747466171734404164956121508, 4.94237668720372598959236413282, 5.79621979815870124946876199695, 6.48262521728980365301557650822, 7.25391515352014540136251503667, 8.013727397903064307471369181752, 8.598322133350903274012974487731, 9.564345278990750227534976181866