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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.426600151\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.426600151\) |
\(L(\frac{3}{2})\) |
|
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
−9.999788561430241389705224689361, −9.015218151971792816674988895054, −7.983899448070284248263130363231, −7.54051759946890412237642288753, −6.55697391089485291786809755441, −6.03005356991666017739724304798, −4.17584906703081557758086079441, −3.58626296875024236992891098431, −3.13102694812244390659670957634, −1.23811256740822211719332792905,
0.64899570627469156207506709468, 2.41808604509201606842948986318, 3.34175857687576406492169744305, 4.27840146116633803654014353865, 5.25991442424400424112940122904, 6.26778377206886052962240234000, 7.18976008193571157496710826530, 8.086776233336451784246516878422, 8.936648937964039955435471673221, 9.258052911207843923640287973381