| L(s) = 1 | + (−3 − 5.19i)11-s − 2·13-s + (−2 + 3.46i)19-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s − 6·29-s + (4 + 6.92i)31-s + (−1 + 1.73i)37-s + 12·41-s − 4·43-s + (−6 + 10.3i)47-s + (−3 − 5.19i)53-s + (−5 + 8.66i)61-s + (−4 − 6.92i)67-s − 6·71-s + ⋯ |
| L(s) = 1 | + (−0.904 − 1.56i)11-s − 0.554·13-s + (−0.458 + 0.794i)19-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s − 1.11·29-s + (0.718 + 1.24i)31-s + (−0.164 + 0.284i)37-s + 1.87·41-s − 0.609·43-s + (−0.875 + 1.51i)47-s + (−0.412 − 0.713i)53-s + (−0.640 + 1.10i)61-s + (−0.488 − 0.846i)67-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4779663785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4779663785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.554574195003549911579618090288, −8.773650660616631093854656284318, −7.960665559929633233705723364521, −7.42644860065159425037676445151, −6.20080554217265571732577823089, −5.64655303408395581646025433378, −4.76355785055075911897485270082, −3.55129892550125268171494229195, −2.85389113381592196300907702039, −1.46082212216262558740401482019,
0.17057248429115436909393366527, 2.07900445752771484411713967524, 2.68751287498481909050364269498, 4.27818700991027018142613131213, 4.69279063319515618283618631482, 5.74946873491688202489108732054, 6.72575848984387405197821086068, 7.43459518678917806391853825802, 8.115740235985354181582674125923, 9.057669002120242481862504988676
