L(s) = 1 | + (0.707 + 0.707i)3-s + (−2.82 + 2.82i)7-s − 1.99i·9-s − 5.19i·11-s + (2.44 − 2.44i)13-s + (3.67 + 3.67i)17-s − 1.73·19-s − 4.00·21-s + (−4.24 − 4.24i)23-s + (3.53 − 3.53i)27-s − 6i·29-s + 3.46i·31-s + (3.67 − 3.67i)33-s + 3.46·39-s − 3·41-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−1.06 + 1.06i)7-s − 0.666i·9-s − 1.56i·11-s + (0.679 − 0.679i)13-s + (0.891 + 0.891i)17-s − 0.397·19-s − 0.872·21-s + (−0.884 − 0.884i)23-s + (0.680 − 0.680i)27-s − 1.11i·29-s + 0.622i·31-s + (0.639 − 0.639i)33-s + 0.554·39-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.490258015\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.490258015\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T + 3iT^{2} \) |
| 7 | \( 1 + (2.82 - 2.82i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 + (-2.44 + 2.44i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.67 - 3.67i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 + (4.24 + 4.24i)T + 23iT^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + (2.82 + 2.82i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.24 + 4.24i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.34 + 7.34i)T - 53iT^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + (4.94 - 4.94i)T - 67iT^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-11.0 + 11.0i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-2.12 - 2.12i)T + 83iT^{2} \) |
| 89 | \( 1 + 3iT - 89T^{2} \) |
| 97 | \( 1 + (-4.89 - 4.89i)T + 97iT^{2} \) |
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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
−9.149497948279362370179134832633, −8.525018119146308159407537539800, −8.149169223597484064351484702628, −6.50774069714977980262300002809, −6.09575636422087888448259561952, −5.43420751861665255395112375650, −3.76040171969889837760772572022, −3.44776938393482798395806279866, −2.44444628650777354346317219911, −0.58003391216163446202421212186,
1.32489099187478995325943930449, 2.42646468810396431444880981899, 3.58936726720667198166722241250, 4.35589024766477088484590008517, 5.41256684913964705008163569532, 6.60572393599272174291924484456, 7.22370706639943134369521671462, 7.65689621599862229183648404916, 8.738627227057794381588840401830, 9.741571187492102807131899793397