L(s) = 1 | + (−0.738 − 0.738i)2-s + (1.99 + 1.99i)3-s − 0.908i·4-s + (0.742 + 2.10i)5-s − 2.94i·6-s + (−0.388 − 0.388i)7-s + (−2.14 + 2.14i)8-s + 4.93i·9-s + (1.01 − 2.10i)10-s + (1.80 − 1.80i)12-s + (−2.29 + 2.29i)13-s + 0.574i·14-s + (−2.72 + 5.67i)15-s + 1.35·16-s + (4.00 + 4.00i)17-s + (3.64 − 3.64i)18-s + ⋯ |
L(s) = 1 | + (−0.522 − 0.522i)2-s + (1.14 + 1.14i)3-s − 0.454i·4-s + (0.331 + 0.943i)5-s − 1.20i·6-s + (−0.146 − 0.146i)7-s + (−0.759 + 0.759i)8-s + 1.64i·9-s + (0.319 − 0.666i)10-s + (0.522 − 0.522i)12-s + (−0.636 + 0.636i)13-s + 0.153i·14-s + (−0.702 + 1.46i)15-s + 0.339·16-s + (0.971 + 0.971i)17-s + (0.858 − 0.858i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31949 + 0.840832i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31949 + 0.840832i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.742 - 2.10i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.738 + 0.738i)T + 2iT^{2} \) |
| 3 | \( 1 + (-1.99 - 1.99i)T + 3iT^{2} \) |
| 7 | \( 1 + (0.388 + 0.388i)T + 7iT^{2} \) |
| 13 | \( 1 + (2.29 - 2.29i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.00 - 4.00i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 + (0.803 + 0.803i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.25T + 29T^{2} \) |
| 31 | \( 1 + 1.64T + 31T^{2} \) |
| 37 | \( 1 + (-0.676 + 0.676i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.94iT - 41T^{2} \) |
| 43 | \( 1 + (2.55 - 2.55i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.87 - 2.87i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.98 - 4.98i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.76iT - 59T^{2} \) |
| 61 | \( 1 - 9.20iT - 61T^{2} \) |
| 67 | \( 1 + (2.62 - 2.62i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.85T + 71T^{2} \) |
| 73 | \( 1 + (-7.13 + 7.13i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.16T + 79T^{2} \) |
| 83 | \( 1 + (-8.01 + 8.01i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.64iT - 89T^{2} \) |
| 97 | \( 1 + (-11.7 + 11.7i)T - 97iT^{2} \) |
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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
−10.25573140962190551019825810143, −10.14386391754014988746529233299, −9.330646352716525453615963185925, −8.571314921992319731443751723305, −7.52368058679721025971083230855, −6.27512756856097496120648589219, −5.15071129557937268781303945940, −3.85800179462073835887980583961, −2.93207976586499789665122512635, −1.96857232272270873946271031174,
0.932631449214235601233510532141, 2.50613717840379311651745221681, 3.43440775999813151499835026093, 5.08406101638046983313401452815, 6.33211004411271679442483425036, 7.31152754456406233485149146948, 7.926978302766921113931944383065, 8.475656357516801243655295404402, 9.373141706304104886960675117231, 9.865332760432092897510080648300