| L(s) = 1 | − i·2-s − 4-s + 3.09i·5-s + (2.44 − i)7-s + i·8-s + 3.09·10-s + (1.79 − 2.79i)11-s + 0.646·13-s + (−1 − 2.44i)14-s + 16-s + 3.74·17-s − 1.80·19-s − 3.09i·20-s + (−2.79 − 1.79i)22-s − 4·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.38i·5-s + (0.925 − 0.377i)7-s + 0.353i·8-s + 0.978·10-s + (0.540 − 0.841i)11-s + 0.179·13-s + (−0.267 − 0.654i)14-s + 0.250·16-s + 0.907·17-s − 0.413·19-s − 0.692i·20-s + (−0.595 − 0.381i)22-s − 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.841446377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.841446377\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.44 + i)T \) |
| 11 | \( 1 + (-1.79 + 2.79i)T \) |
| good | 5 | \( 1 - 3.09iT - 5T^{2} \) |
| 13 | \( 1 - 0.646T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 1.58iT - 29T^{2} \) |
| 31 | \( 1 - 8.64iT - 31T^{2} \) |
| 37 | \( 1 - 3.58T + 37T^{2} \) |
| 41 | \( 1 - 9.93T + 41T^{2} \) |
| 43 | \( 1 - 7.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.93iT - 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 0.646iT - 59T^{2} \) |
| 61 | \( 1 - 1.93T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 - 8.50iT - 97T^{2} \) |
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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
−9.830026426971797215981268325377, −8.736266638810156945229863180902, −8.017902718413666291110799889173, −7.16347543025785488377431406072, −6.26843143484428845652848425835, −5.35414760262609230872059157172, −4.11202932288474047138367090809, −3.40794353248715910125853897215, −2.42626054674739687636906035675, −1.16096228776303297672964214644,
0.969545393288141126297892176315, 2.13687339910709612899241854598, 4.10652268999401654993347048353, 4.51273341886007975622237289294, 5.52095434584704191252006447440, 6.05849365707748282156146407547, 7.44409209947828515942336943122, 7.952108066750676008191235048685, 8.731440761961205795546182828538, 9.343420806576129491358655304142
