L(s) = 1 | + (0.707 + 0.707i)2-s + (1.73 + 0.0444i)3-s + 1.00i·4-s + (0.142 − 2.23i)5-s + (1.19 + 1.25i)6-s + (2.34 − 2.34i)7-s + (−0.707 + 0.707i)8-s + (2.99 + 0.153i)9-s + (1.67 − 1.47i)10-s + 5.32i·11-s + (−0.0444 + 1.73i)12-s + (−2.57 − 2.57i)13-s + 3.31·14-s + (0.345 − 3.85i)15-s − 1.00·16-s + (2.33 + 2.33i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.999 + 0.0256i)3-s + 0.500i·4-s + (0.0635 − 0.997i)5-s + (0.487 + 0.512i)6-s + (0.885 − 0.885i)7-s + (−0.250 + 0.250i)8-s + (0.998 + 0.0512i)9-s + (0.530 − 0.467i)10-s + 1.60i·11-s + (−0.0128 + 0.499i)12-s + (−0.714 − 0.714i)13-s + 0.885·14-s + (0.0891 − 0.996i)15-s − 0.250·16-s + (0.565 + 0.565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.15298 + 0.225091i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.15298 + 0.225091i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.73 - 0.0444i)T \) |
| 5 | \( 1 + (-0.142 + 2.23i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + (-2.34 + 2.34i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.32iT - 11T^{2} \) |
| 13 | \( 1 + (2.57 + 2.57i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.33 - 2.33i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.87iT - 19T^{2} \) |
| 23 | \( 1 + (2.05 - 2.05i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 37 | \( 1 + (-0.718 + 0.718i)T - 37iT^{2} \) |
| 41 | \( 1 + 9.62iT - 41T^{2} \) |
| 43 | \( 1 + (-1.71 - 1.71i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.49 - 6.49i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.55 - 5.55i)T - 53iT^{2} \) |
| 59 | \( 1 + 6.86T + 59T^{2} \) |
| 61 | \( 1 + 0.913T + 61T^{2} \) |
| 67 | \( 1 + (9.71 - 9.71i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.59iT - 71T^{2} \) |
| 73 | \( 1 + (5.80 + 5.80i)T + 73iT^{2} \) |
| 79 | \( 1 - 10.0iT - 79T^{2} \) |
| 83 | \( 1 + (4.35 - 4.35i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.15T + 89T^{2} \) |
| 97 | \( 1 + (7.10 - 7.10i)T - 97iT^{2} \) |
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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
−9.889373118396265317558705023817, −9.154548569257200190179615306213, −8.176345184338689202656306292478, −7.57742988435124230564812084160, −7.06351129759900836476128753285, −5.46861407324701283313509455132, −4.48261353878351139699736022564, −4.26376530240167239541139640026, −2.65545659559590415387867340472, −1.40900534640446866025593633050,
1.71014034433388636229320403678, 2.70553060101944157748385982320, 3.38362560541089456651844354437, 4.52533570251799899678591048455, 5.68999715615854525271179344214, 6.51209782986654298606610009294, 7.73261721125192979884911165620, 8.340959877875778179004961528911, 9.278716465860502668132768575488, 10.11162995318009735750150970004