L(s) = 1 | + 3-s + (1.98 + 1.98i)5-s + (3.05 − 3.05i)7-s + 9-s + (1.08 + 1.08i)11-s + (−3.57 − 0.429i)13-s + (1.98 + 1.98i)15-s − 5.70i·17-s + (4.39 − 4.39i)19-s + (3.05 − 3.05i)21-s + 2.95·23-s + 2.84i·25-s + 27-s + 6.96i·29-s + (−3.05 − 3.05i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (0.885 + 0.885i)5-s + (1.15 − 1.15i)7-s + 0.333·9-s + (0.327 + 0.327i)11-s + (−0.992 − 0.119i)13-s + (0.511 + 0.511i)15-s − 1.38i·17-s + (1.00 − 1.00i)19-s + (0.667 − 0.667i)21-s + 0.615·23-s + 0.569i·25-s + 0.192·27-s + 1.29i·29-s + (−0.549 − 0.549i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.695556939\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.695556939\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (3.57 + 0.429i)T \) |
good | 5 | \( 1 + (-1.98 - 1.98i)T + 5iT^{2} \) |
| 7 | \( 1 + (-3.05 + 3.05i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.08 - 1.08i)T + 11iT^{2} \) |
| 17 | \( 1 + 5.70iT - 17T^{2} \) |
| 19 | \( 1 + (-4.39 + 4.39i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.95T + 23T^{2} \) |
| 29 | \( 1 - 6.96iT - 29T^{2} \) |
| 31 | \( 1 + (3.05 + 3.05i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.14 - 5.14i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.77 - 2.77i)T - 41iT^{2} \) |
| 43 | \( 1 + 3.00iT - 43T^{2} \) |
| 47 | \( 1 + (6.40 - 6.40i)T - 47iT^{2} \) |
| 53 | \( 1 - 4.28iT - 53T^{2} \) |
| 59 | \( 1 + (3.00 + 3.00i)T + 59iT^{2} \) |
| 61 | \( 1 + 1.13iT - 61T^{2} \) |
| 67 | \( 1 + (7.39 - 7.39i)T - 67iT^{2} \) |
| 71 | \( 1 + (-3.20 - 3.20i)T + 71iT^{2} \) |
| 73 | \( 1 + (-7.87 - 7.87i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.46iT - 79T^{2} \) |
| 83 | \( 1 + (-9.78 + 9.78i)T - 83iT^{2} \) |
| 89 | \( 1 + (-4.24 - 4.24i)T + 89iT^{2} \) |
| 97 | \( 1 + (11.5 - 11.5i)T - 97iT^{2} \) |
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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
−9.692564659652811733040895579782, −9.071721146179939481771643336062, −7.82439730457682644171906613874, −7.14896594158160238163586379947, −6.81163741405332141881058130210, −5.16491228383053170876021382826, −4.71606106332859354258641869232, −3.30866010936863580495402822429, −2.45573451549901257486702449542, −1.26979577037839984271566684327,
1.58162946526905246480018974774, 2.09693664554322259915750725085, 3.52574633322146833311237546260, 4.81836727987360986712797462092, 5.39965394707777385118866237005, 6.15819731892827321680762614910, 7.53631652177494933318023338631, 8.298231110726326819622321696936, 8.875492138365261622977041149724, 9.507868568823121994473397049657