L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.30 − 2.26i)5-s + (2.56 + 4.44i)7-s + 0.999·8-s − 2.61·10-s + (−1.23 − 2.14i)11-s + (−1.90 + 3.29i)13-s + (2.56 − 4.44i)14-s + (−0.5 − 0.866i)16-s + 2.41·17-s + 5.05·19-s + (1.30 + 2.26i)20-s + (−1.23 + 2.14i)22-s + (−2.85 + 4.95i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.583 − 1.01i)5-s + (0.970 + 1.68i)7-s + 0.353·8-s − 0.825·10-s + (−0.373 − 0.646i)11-s + (−0.527 + 0.912i)13-s + (0.686 − 1.18i)14-s + (−0.125 − 0.216i)16-s + 0.584·17-s + 1.16·19-s + (0.291 + 0.505i)20-s + (−0.264 + 0.457i)22-s + (−0.596 + 1.03i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.634973411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.634973411\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.30 + 2.26i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.56 - 4.44i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.23 + 2.14i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.90 - 3.29i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 + (2.85 - 4.95i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.81 - 3.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.01 + 3.48i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.49T + 37T^{2} \) |
| 41 | \( 1 + (5.17 - 8.95i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.128 - 0.222i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.62 + 6.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 + (1.71 - 2.96i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.80 - 8.32i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.63 + 2.82i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 + (0.482 + 0.835i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.20 + 3.82i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6.51T + 89T^{2} \) |
| 97 | \( 1 + (4.64 + 8.05i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.486890149161901905220492296546, −8.731046323867989289374617469082, −8.335879549446603276830383864658, −7.37213911933591070869526238638, −5.84169377662262858813665841055, −5.34234923101011666440817717600, −4.64951838941897042262854038611, −3.17045577224255383394893883945, −2.11106882218726426016184895999, −1.31635118850395789101004920857,
0.830951549510869611314560700481, 2.22503923468128261783554554033, 3.50978437933827345662445681396, 4.66962370864727999493567102522, 5.37389393544262362367956994886, 6.50907168958055422300368750632, 7.26999528424176170409875787482, 7.65011645404297052753597729251, 8.444637957942088226785145225040, 9.906847312180315887306439452835