L(s) = 1 | + (−2.82 − i)3-s + 7i·7-s + (7.00 + 5.65i)9-s + 8.48i·11-s + 25i·13-s − 25.4·17-s − 7·19-s + (7 − 19.7i)21-s + 25.4·23-s + (−14.1 − 23.0i)27-s − 42.4i·29-s + 7·31-s + (8.48 − 24i)33-s + 2i·37-s + (25 − 70.7i)39-s + ⋯ |
L(s) = 1 | + (−0.942 − 0.333i)3-s + i·7-s + (0.777 + 0.628i)9-s + 0.771i·11-s + 1.92i·13-s − 1.49·17-s − 0.368·19-s + (0.333 − 0.942i)21-s + 1.10·23-s + (−0.523 − 0.851i)27-s − 1.46i·29-s + 0.225·31-s + (0.257 − 0.727i)33-s + 0.0540i·37-s + (0.641 − 1.81i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4173875947\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4173875947\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.82 + i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7iT - 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 25iT - 169T^{2} \) |
| 17 | \( 1 + 25.4T + 289T^{2} \) |
| 19 | \( 1 + 7T + 361T^{2} \) |
| 23 | \( 1 - 25.4T + 529T^{2} \) |
| 29 | \( 1 + 42.4iT - 841T^{2} \) |
| 31 | \( 1 - 7T + 961T^{2} \) |
| 37 | \( 1 - 2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.48iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 41iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 59.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + T + 3.72e3T^{2} \) |
| 67 | \( 1 + 17iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 42.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 58T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 49iT - 9.40e3T^{2} \) |
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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
−9.868510523033349492411751989661, −9.243392408689948481495068316053, −8.437091446325042243470049496563, −7.21982102644315627328214950739, −6.60181790402161353176180922733, −5.97069480624930357531892700891, −4.71503810520521658471173896153, −4.36684393713173736132082266467, −2.43831101400054183109312140101, −1.68771663962091490547487821082,
0.15869151030598197081948981726, 1.09540097078579645140606667992, 2.99214501888741818460519181355, 3.94755524780245079122928896907, 4.93108165515198497351579105964, 5.65011818765130617805514609886, 6.64278964451902735253415329976, 7.27723343523833493330386068131, 8.352634141174536082813198224479, 9.181341652313444350595862525179