L(s) = 1 | + (1.72 + 1.72i)2-s + (−2.01 − 2.01i)3-s + 3.95i·4-s − 6.93i·6-s + (−0.725 − 0.725i)7-s + (−3.37 + 3.37i)8-s + 5.08i·9-s + 4.60i·11-s + (7.95 − 7.95i)12-s + (−1.16 − 1.16i)13-s − 2.50i·14-s − 3.74·16-s + (−3.76 + 3.76i)17-s + (−8.77 + 8.77i)18-s + 3.87i·19-s + ⋯ |
L(s) = 1 | + (1.22 + 1.22i)2-s + (−1.16 − 1.16i)3-s + 1.97i·4-s − 2.83i·6-s + (−0.274 − 0.274i)7-s + (−1.19 + 1.19i)8-s + 1.69i·9-s + 1.38i·11-s + (2.29 − 2.29i)12-s + (−0.323 − 0.323i)13-s − 0.669i·14-s − 0.936·16-s + (−0.912 + 0.912i)17-s + (−2.06 + 2.06i)18-s + 0.888i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.461428 + 1.34818i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.461428 + 1.34818i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 31 | \( 1 + (5.53 + 0.588i)T \) |
good | 2 | \( 1 + (-1.72 - 1.72i)T + 2iT^{2} \) |
| 3 | \( 1 + (2.01 + 2.01i)T + 3iT^{2} \) |
| 7 | \( 1 + (0.725 + 0.725i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.60iT - 11T^{2} \) |
| 13 | \( 1 + (1.16 + 1.16i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.76 - 3.76i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.87iT - 19T^{2} \) |
| 23 | \( 1 + (-5.77 - 5.77i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.95T + 29T^{2} \) |
| 37 | \( 1 + (4.78 - 4.78i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.44T + 41T^{2} \) |
| 43 | \( 1 + (2.15 + 2.15i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.45 + 1.45i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.61 + 3.61i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.09iT - 59T^{2} \) |
| 61 | \( 1 + 8.38iT - 61T^{2} \) |
| 67 | \( 1 + (-7.01 - 7.01i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.76T + 71T^{2} \) |
| 73 | \( 1 + (1.92 + 1.92i)T + 73iT^{2} \) |
| 79 | \( 1 - 2.57T + 79T^{2} \) |
| 83 | \( 1 + (-1.48 - 1.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + (-4.21 - 4.21i)T + 97iT^{2} \) |
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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
−10.92992989051053711699657347012, −9.914232046811880893606112113072, −8.366443592908467994676509511549, −7.44873494427393139959942614475, −6.92332231329647475379927049512, −6.39143181583565992892761856868, −5.39334586428436923860496114442, −4.82475905344627039625521514065, −3.59722554719558451645270045594, −1.78947777798892599928802488631,
0.55018195508945231827894783702, 2.64718216696052170010239602105, 3.51882154693927235120081750628, 4.66615896167522996520367175450, 5.01131564348267371432198326452, 5.98032209453777258869504576453, 6.75436155346191937152371672693, 8.847565062074216363542976786160, 9.459452418494332015139080402731, 10.56753913511970975985547128863