L(s) = 1 | + (1.80 + 1.04i)2-s + (−0.886 + 1.48i)3-s + (1.17 + 2.03i)4-s + (1.65 + 2.86i)5-s + (−3.15 + 1.76i)6-s + 0.717i·8-s + (−1.42 − 2.63i)9-s + 6.88i·10-s + (2.30 + 1.33i)11-s + (−4.06 − 0.0560i)12-s + (2.11 − 1.21i)13-s + (−5.72 − 0.0790i)15-s + (1.59 − 2.76i)16-s − 7.18·17-s + (0.172 − 6.25i)18-s − 4.90i·19-s + ⋯ |
L(s) = 1 | + (1.27 + 0.736i)2-s + (−0.511 + 0.859i)3-s + (0.586 + 1.01i)4-s + (0.738 + 1.27i)5-s + (−1.28 + 0.719i)6-s + 0.253i·8-s + (−0.475 − 0.879i)9-s + 2.17i·10-s + (0.694 + 0.401i)11-s + (−1.17 − 0.0161i)12-s + (0.585 − 0.338i)13-s + (−1.47 − 0.0203i)15-s + (0.399 − 0.691i)16-s − 1.74·17-s + (0.0406 − 1.47i)18-s − 1.12i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08000 + 2.28533i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08000 + 2.28533i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.886 - 1.48i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.80 - 1.04i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.65 - 2.86i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.30 - 1.33i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.11 + 1.21i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 7.18T + 17T^{2} \) |
| 19 | \( 1 + 4.90iT - 19T^{2} \) |
| 23 | \( 1 + (4.32 - 2.49i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.50 - 3.17i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.30 + 1.33i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 + (-0.553 - 0.958i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.93 + 5.08i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.44 + 4.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (2.56 + 4.44i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.44 - 2.56i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.16 + 7.21i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.07iT - 71T^{2} \) |
| 73 | \( 1 - 8.01iT - 73T^{2} \) |
| 79 | \( 1 + (2.50 - 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.04 + 1.80i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 + (9.47 + 5.46i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.40179790563166459119302696488, −10.68252812648491382567736735100, −9.826916233711997233790319299699, −8.848552061643552872567611658472, −7.06140565258689041760449632936, −6.50263112693909709011198496528, −5.85933612527096703328871958730, −4.72739138855235072242342004212, −3.86251380081593313613102511283, −2.70784809517187901366605552774,
1.33560881421818745897919091249, 2.30069933280969532452992044971, 4.09449412250511175222165084993, 4.88903302078456494570568819882, 5.99762200893921442716357150600, 6.39690735223345973806561215096, 8.226642239787001990732848973733, 8.871206901088161933862752623232, 10.25265336560546524276213623449, 11.26564966475118900035077804434