L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.10 + 1.33i)3-s + 1.00i·4-s + (−0.953 − 2.02i)5-s + (1.72 − 0.162i)6-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.559 − 2.94i)9-s + (−0.755 + 2.10i)10-s − 0.780i·11-s + (−1.33 − 1.10i)12-s + (−3.85 − 3.85i)13-s + 1.00·14-s + (3.75 + 0.961i)15-s − 1.00·16-s + (−2.97 − 2.97i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.637 + 0.770i)3-s + 0.500i·4-s + (−0.426 − 0.904i)5-s + (0.703 − 0.0662i)6-s + (−0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.186 − 0.982i)9-s + (−0.238 + 0.665i)10-s − 0.235i·11-s + (−0.385 − 0.318i)12-s + (−1.06 − 1.06i)13-s + 0.267·14-s + (0.968 + 0.248i)15-s − 0.250·16-s + (−0.721 − 0.721i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.134597 - 0.333866i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.134597 - 0.333866i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.10 - 1.33i)T \) |
| 5 | \( 1 + (0.953 + 2.02i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 + 0.780iT - 11T^{2} \) |
| 13 | \( 1 + (3.85 + 3.85i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.97 + 2.97i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.79iT - 19T^{2} \) |
| 23 | \( 1 + (1.74 - 1.74i)T - 23iT^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 - 1.20T + 31T^{2} \) |
| 37 | \( 1 + (6.28 - 6.28i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.410iT - 41T^{2} \) |
| 43 | \( 1 + (0.397 + 0.397i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.50 - 4.50i)T + 47iT^{2} \) |
| 53 | \( 1 + (-7.69 + 7.69i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 9.11T + 61T^{2} \) |
| 67 | \( 1 + (-4.95 + 4.95i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.88iT - 71T^{2} \) |
| 73 | \( 1 + (8.48 + 8.48i)T + 73iT^{2} \) |
| 79 | \( 1 - 12.5iT - 79T^{2} \) |
| 83 | \( 1 + (-3.60 + 3.60i)T - 83iT^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 + (8.82 - 8.82i)T - 97iT^{2} \) |
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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
−11.90007019240768038724963942051, −11.07098495831927629394445082101, −9.989200185355156494962902323531, −9.241654931437782978182734384919, −8.339327652959152339923373220697, −6.96900669359268062046848588839, −5.35896347007874950537468761361, −4.51813159691538709770590846214, −2.99063287127576143716061488326, −0.37338600230011037618993655286,
2.13312028394773834110502183153, 4.27119608773980869787695386903, 5.89121887026802407590415931944, 6.83749268736496952139700374403, 7.39563516372366711298357848160, 8.509042515339502024236030966603, 10.02323116303156404457135562699, 10.68953391390665619145598867036, 11.79444630925261988947620876683, 12.49928111354663423888305546630