L(s) = 1 | + (1.56 − 2.70i)5-s + (2.37 + 1.17i)7-s + (3.23 − 1.87i)11-s + 2.93i·13-s + (2.71 + 4.70i)17-s + (−6.07 − 3.50i)19-s + (−1.44 − 0.832i)23-s + (−2.37 − 4.10i)25-s − 7.07i·29-s + (6.60 − 3.81i)31-s + (6.87 − 4.57i)35-s + (−4.03 + 6.98i)37-s + 9.60·41-s − 5.74·43-s + (1.56 − 2.70i)47-s + ⋯ |
L(s) = 1 | + (0.697 − 1.20i)5-s + (0.895 + 0.444i)7-s + (0.976 − 0.563i)11-s + 0.812i·13-s + (0.658 + 1.14i)17-s + (−1.39 − 0.804i)19-s + (−0.300 − 0.173i)23-s + (−0.474 − 0.821i)25-s − 1.31i·29-s + (1.18 − 0.685i)31-s + (1.16 − 0.773i)35-s + (−0.663 + 1.14i)37-s + 1.49·41-s − 0.875·43-s + (0.227 − 0.394i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.88416 - 0.569422i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88416 - 0.569422i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.37 - 1.17i)T \) |
good | 5 | \( 1 + (-1.56 + 2.70i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.23 + 1.87i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 + (-2.71 - 4.70i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.07 + 3.50i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.44 + 0.832i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.07iT - 29T^{2} \) |
| 31 | \( 1 + (-6.60 + 3.81i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.03 - 6.98i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.60T + 41T^{2} \) |
| 43 | \( 1 + 5.74T + 43T^{2} \) |
| 47 | \( 1 + (-1.56 + 2.70i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.11 - 1.79i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.20 + 3.81i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.57 + 1.48i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.90 - 8.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.33iT - 71T^{2} \) |
| 73 | \( 1 + (-2.46 + 1.42i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.57 - 4.45i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + (-2.59 + 4.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.39iT - 97T^{2} \) |
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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
−10.13495650201694718808343969588, −9.234633130988771557749930673605, −8.590636162437739416950665444068, −8.066411738866566524789583788741, −6.45056854162401002763317952973, −5.89656483838328032582453526350, −4.74428435127236175485680379300, −4.10242690499596127459282104278, −2.21158311085030236538801099781, −1.23827794568111362717939462634,
1.52411431677430727389562670707, 2.73507863402643284123443737634, 3.91067999141104340574854238629, 5.06746695494155694985718881715, 6.12582561143418168427643535094, 6.95739215736572525994737269857, 7.67081140994425536217531462215, 8.728384231470793360602856602063, 9.809871061766522927713542673405, 10.44816494764851221685249636768