L(s) = 1 | − 0.512·2-s − 3.19i·3-s − 1.73·4-s − 5-s + 1.64i·6-s − 1.77i·7-s + 1.91·8-s − 7.23·9-s + 0.512·10-s + (−0.881 − 3.19i)11-s + 5.55i·12-s − 2.41·13-s + 0.909i·14-s + 3.19i·15-s + 2.49·16-s + 1.90i·17-s + ⋯ |
L(s) = 1 | − 0.362·2-s − 1.84i·3-s − 0.868·4-s − 0.447·5-s + 0.669i·6-s − 0.669i·7-s + 0.677·8-s − 2.41·9-s + 0.162·10-s + (−0.265 − 0.964i)11-s + 1.60i·12-s − 0.669·13-s + 0.242i·14-s + 0.825i·15-s + 0.622·16-s + 0.462i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2100042840\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2100042840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 + (0.881 + 3.19i)T \) |
| 19 | \( 1 + (3.68 + 2.33i)T \) |
good | 2 | \( 1 + 0.512T + 2T^{2} \) |
| 3 | \( 1 + 3.19iT - 3T^{2} \) |
| 7 | \( 1 + 1.77iT - 7T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 - 1.90iT - 17T^{2} \) |
| 23 | \( 1 + 3.67T + 23T^{2} \) |
| 29 | \( 1 - 7.99T + 29T^{2} \) |
| 31 | \( 1 + 0.955iT - 31T^{2} \) |
| 37 | \( 1 - 6.90iT - 37T^{2} \) |
| 41 | \( 1 + 0.482T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 - 0.519T + 47T^{2} \) |
| 53 | \( 1 - 2.92iT - 53T^{2} \) |
| 59 | \( 1 + 10.0iT - 59T^{2} \) |
| 61 | \( 1 - 7.83iT - 61T^{2} \) |
| 67 | \( 1 - 3.84iT - 67T^{2} \) |
| 71 | \( 1 - 0.976iT - 71T^{2} \) |
| 73 | \( 1 + 0.0771iT - 73T^{2} \) |
| 79 | \( 1 - 7.50T + 79T^{2} \) |
| 83 | \( 1 + 2.81iT - 83T^{2} \) |
| 89 | \( 1 - 15.4iT - 89T^{2} \) |
| 97 | \( 1 - 6.02iT - 97T^{2} \) |
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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
−8.785633027962289408380451988584, −8.336645272009750595005453336403, −7.71223953200162379782636696310, −6.92385603783517039718330968233, −6.10100515261738852500645785805, −4.98803442018069421562441382909, −3.77124225056410753527857698361, −2.47600755437238319234194000808, −1.05476860900071628158541068316, −0.13195723738344920004560210492,
2.53414824929820889913190111825, 3.75751254133122662481284663612, 4.59246160164912481123503238722, 4.96987258479477647241671249057, 6.04877619061907973540017612391, 7.62178980989138127979600257732, 8.448769702738850008080519532318, 9.067356547404341570492078559236, 9.849348010091754271520893927603, 10.18093131276336199542738609452