L(s) = 1 | + (0.5 − 0.866i)5-s + (−0.724 − 1.25i)7-s + (−1.72 − 2.98i)11-s + (−1.94 + 3.37i)13-s + 4.89·17-s + 4·19-s + (0.275 − 0.476i)23-s + (2 + 3.46i)25-s + (−4.94 − 8.57i)29-s + (3.72 − 6.45i)31-s − 1.44·35-s − 8.89·37-s + (−1.05 + 1.81i)41-s + (−6.17 − 10.6i)43-s + (−4.17 − 7.22i)47-s + ⋯ |
L(s) = 1 | + (0.223 − 0.387i)5-s + (−0.273 − 0.474i)7-s + (−0.520 − 0.900i)11-s + (−0.540 + 0.936i)13-s + 1.18·17-s + 0.917·19-s + (0.0573 − 0.0994i)23-s + (0.400 + 0.692i)25-s + (−0.919 − 1.59i)29-s + (0.668 − 1.15i)31-s − 0.245·35-s − 1.46·37-s + (−0.164 + 0.284i)41-s + (−0.941 − 1.63i)43-s + (−0.608 − 1.05i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.299499047\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.299499047\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.724 + 1.25i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.72 + 2.98i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.94 - 3.37i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-0.275 + 0.476i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.94 + 8.57i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.72 + 6.45i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.89T + 37T^{2} \) |
| 41 | \( 1 + (1.05 - 1.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.17 + 10.6i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.17 + 7.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 0.898T + 53T^{2} \) |
| 59 | \( 1 + (0.174 - 0.301i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.949 - 1.64i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.17 + 2.03i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 + (-4.27 - 7.40i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.72 + 4.71i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 + (2.94 + 5.10i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.167503392405885850322441795273, −8.285893549969246733167744471909, −7.51598061910138077634814658303, −6.77664621591988753179092113809, −5.67874807479054002040810889853, −5.16626778037866505120404592816, −3.96865793867901402036772110989, −3.16850317437070569483885122609, −1.87455573377407387250819280943, −0.49534822018451821374829266810,
1.45016841395675930628219374714, 2.82466536515815074244255905129, 3.31052914307582944325053660132, 4.93418879114469726116586135294, 5.31857532502498323885209081268, 6.37251928689621295388751361648, 7.26724101378382294570094138658, 7.82866371403245292611585420724, 8.802746930339380368089226030635, 9.730725844366548633180933332948