L(s) = 1 | − i·2-s + (1.73 + 0.0164i)3-s − 4-s + 3.75·5-s + (0.0164 − 1.73i)6-s − i·7-s + i·8-s + (2.99 + 0.0568i)9-s − 3.75i·10-s + 2.91·11-s + (−1.73 − 0.0164i)12-s − 1.96·13-s − 14-s + (6.51 + 0.0616i)15-s + 16-s − 6.79·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.999 + 0.00946i)3-s − 0.5·4-s + 1.68·5-s + (0.00669 − 0.707i)6-s − 0.377i·7-s + 0.353i·8-s + (0.999 + 0.0189i)9-s − 1.18i·10-s + 0.878·11-s + (−0.499 − 0.00473i)12-s − 0.544·13-s − 0.267·14-s + (1.68 + 0.0159i)15-s + 0.250·16-s − 1.64·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.47034 - 1.41046i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.47034 - 1.41046i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.73 - 0.0164i)T \) |
| 7 | \( 1 + iT \) |
| 23 | \( 1 + (-2.47 - 4.10i)T \) |
good | 5 | \( 1 - 3.75T + 5T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 + 1.96T + 13T^{2} \) |
| 17 | \( 1 + 6.79T + 17T^{2} \) |
| 19 | \( 1 + 5.60iT - 19T^{2} \) |
| 29 | \( 1 - 9.69iT - 29T^{2} \) |
| 31 | \( 1 + 8.11T + 31T^{2} \) |
| 37 | \( 1 + 1.15iT - 37T^{2} \) |
| 41 | \( 1 + 1.92iT - 41T^{2} \) |
| 43 | \( 1 + 6.91iT - 43T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 + 6.79iT - 59T^{2} \) |
| 61 | \( 1 + 5.64iT - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 4.18iT - 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 8.47iT - 79T^{2} \) |
| 83 | \( 1 - 8.52T + 83T^{2} \) |
| 89 | \( 1 - 5.22T + 89T^{2} \) |
| 97 | \( 1 - 1.81iT - 97T^{2} \) |
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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
−9.645735602580092539676134615781, −9.129648948771966817655028204824, −8.839508734305136407139115613039, −7.20673077286033042362671671096, −6.68245636764203978141542752242, −5.31241012907853311323782379304, −4.43997383479680169603983921686, −3.23518432958325979652151092793, −2.24454352221262107860650460829, −1.47766752252516458226478784625,
1.72961777252473366751061241877, 2.53170012966595967839404223702, 3.99898515346771848757409099034, 4.98498245965406290465058495522, 6.14671862024263659591622782387, 6.57866493211712368291881074404, 7.65507107039938795023120151029, 8.744111016684190245773157314856, 9.153058130824630403724785038106, 9.818238285168696957377398056110