L(s) = 1 | − i·2-s + 3-s − 4-s + i·5-s − i·6-s − 5.04i·7-s + i·8-s + 9-s + 10-s + 4.74i·11-s − 12-s − 5.04·14-s + i·15-s + 16-s + 1.08·17-s − i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 1.90i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.42i·11-s − 0.288·12-s − 1.34·14-s + 0.258i·15-s + 0.250·16-s + 0.263·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9239325254\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9239325254\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 5.04iT - 7T^{2} \) |
| 11 | \( 1 - 4.74iT - 11T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 - 4.44iT - 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 7.85iT - 31T^{2} \) |
| 37 | \( 1 - 7.14iT - 37T^{2} \) |
| 41 | \( 1 + 0.664iT - 41T^{2} \) |
| 43 | \( 1 + 4.35T + 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 12.2iT - 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 9.58iT - 67T^{2} \) |
| 71 | \( 1 - 4.47iT - 71T^{2} \) |
| 73 | \( 1 - 8.17iT - 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 4.86iT - 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−8.201936389305610556962105558442, −7.72403662437277014387777453097, −7.08416527402239686696743844122, −6.50383477039826065450740011835, −5.15248734375405740679010275781, −4.41548713702403316099053661059, −3.72960354445876380533247773354, −3.25507157292930185346060352037, −1.93807948021323121121811677066, −1.37162365133848167514726762271,
0.21635385889549210970610749489, 1.79362052713613289294613167057, 2.73594018512066361243243007140, 3.48084904034790716860825364281, 4.52864408109536303777072292283, 5.40102050473992911365263708573, 5.89233423771178234828580302430, 6.40787415542506610325541214977, 7.74576771711430475601883530639, 7.977178589807757351228986094272