| L(s) = 1 | − i·2-s − 4-s − 2.97·5-s + (2.55 − 0.698i)7-s + i·8-s + 2.97i·10-s − i·11-s + 1.97i·13-s + (−0.698 − 2.55i)14-s + 16-s + 1.57·17-s + 7.18i·19-s + 2.97·20-s − 22-s + 3.86·25-s + 1.97·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.33·5-s + (0.964 − 0.264i)7-s + 0.353i·8-s + 0.941i·10-s − 0.301i·11-s + 0.548i·13-s + (−0.186 − 0.681i)14-s + 0.250·16-s + 0.382·17-s + 1.64i·19-s + 0.665·20-s − 0.213·22-s + 0.772·25-s + 0.387·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.303012201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.303012201\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.55 + 0.698i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.97T + 5T^{2} \) |
| 13 | \( 1 - 1.97iT - 13T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 - 7.18iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 1.30iT - 29T^{2} \) |
| 31 | \( 1 + 5.78iT - 31T^{2} \) |
| 37 | \( 1 - 8.04T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 - 9.14T + 43T^{2} \) |
| 47 | \( 1 - 2.63T + 47T^{2} \) |
| 53 | \( 1 + 8.94iT - 53T^{2} \) |
| 59 | \( 1 - 9.90T + 59T^{2} \) |
| 61 | \( 1 - 4.77iT - 61T^{2} \) |
| 67 | \( 1 - 0.699T + 67T^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 - 8.20T + 89T^{2} \) |
| 97 | \( 1 + 8.41iT - 97T^{2} \) |
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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
−9.570311772450825437313969254322, −8.554124115314355816233149205666, −7.926940527860254664957157165307, −7.45022376435390490869189543380, −6.08833528396123630216288411246, −5.02098013485731711029570124874, −4.05271755393519355226235944752, −3.65893419759960078083003229197, −2.18850716716216144721788529402, −0.896942950603761679239628991824,
0.797478043415098900186180887723, 2.64964281012363800557508065953, 3.88421133666752189858449064454, 4.69449198849110153136387088294, 5.34672170445705546438347805135, 6.54929209722813632257326672928, 7.49701043059899836864636268968, 7.84378550870892044339706119681, 8.656436248365491310722147293799, 9.354656330059919872494729729579
