L(s) = 1 | + (−2.36 − 1.36i)2-s + (−0.577 − 1.00i)3-s + (2.73 + 4.74i)4-s + (−1.62 − 0.938i)5-s + 3.16i·6-s − 9.50i·8-s + (0.831 − 1.44i)9-s + (2.56 + 4.44i)10-s + (1.99 − 1.14i)11-s + (3.16 − 5.48i)12-s + (−0.574 + 3.55i)13-s + 2.17i·15-s + (−7.51 + 13.0i)16-s + (3.03 + 5.25i)17-s + (−3.93 + 2.27i)18-s + (4.46 + 2.57i)19-s + ⋯ |
L(s) = 1 | + (−1.67 − 0.966i)2-s + (−0.333 − 0.577i)3-s + (1.36 + 2.37i)4-s + (−0.727 − 0.419i)5-s + 1.29i·6-s − 3.36i·8-s + (0.277 − 0.480i)9-s + (0.811 + 1.40i)10-s + (0.600 − 0.346i)11-s + (0.913 − 1.58i)12-s + (−0.159 + 0.987i)13-s + 0.560i·15-s + (−1.87 + 3.25i)16-s + (0.736 + 1.27i)17-s + (−0.928 + 0.536i)18-s + (1.02 + 0.590i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.280357 - 0.464210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.280357 - 0.464210i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (0.574 - 3.55i)T \) |
good | 2 | \( 1 + (2.36 + 1.36i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.577 + 1.00i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.62 + 0.938i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.99 + 1.14i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.03 - 5.25i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.46 - 2.57i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.20 + 3.82i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.50T + 29T^{2} \) |
| 31 | \( 1 + (-3.75 + 2.16i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.74 - 1.58i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.45iT - 41T^{2} \) |
| 43 | \( 1 + 2.17T + 43T^{2} \) |
| 47 | \( 1 + (7.71 + 4.45i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.05 + 7.01i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.18 + 3.56i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.00 + 6.93i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.00 + 0.578i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.11iT - 71T^{2} \) |
| 73 | \( 1 + (-1.70 + 0.984i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.90 - 3.30i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.49iT - 83T^{2} \) |
| 89 | \( 1 + (9.03 + 5.21i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.51iT - 97T^{2} \) |
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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
−10.06914556830247106224782390268, −9.614275132961937258211653172838, −8.399065416486652993826634449413, −8.158964527477429401445843620392, −6.98485455296138399741590367180, −6.35640754081658458156382145765, −4.21371522127191066525001828120, −3.30048944884010114291005645980, −1.67673509261718450299256919859, −0.74772750844397244072916838435,
1.01590887815568959790693404197, 2.95046186632033927310804779787, 4.83884296780615919308149447431, 5.57635028831499698156271149748, 6.87238311977662531032811130089, 7.47521335612836806934543969026, 8.052857482986045006518371117541, 9.271839818251439632564700904531, 9.825153429955709772506414627053, 10.51931967707084066525349847374