L(s) = 1 | + (0.896 + 0.517i)5-s + (0.232 − 0.133i)7-s + (−1.93 − 3.34i)11-s + (1.23 − 2.13i)13-s − 6.69i·17-s + 1.73i·19-s + (−2.96 + 5.13i)23-s + (−1.96 − 3.40i)25-s + (−1.79 + 1.03i)29-s + (0.464 + 0.267i)31-s + 0.277·35-s − 6.46·37-s + (1.79 + 1.03i)41-s + (−6.46 + 3.73i)43-s + (−4.76 − 8.24i)47-s + ⋯ |
L(s) = 1 | + (0.400 + 0.231i)5-s + (0.0877 − 0.0506i)7-s + (−0.582 − 1.00i)11-s + (0.341 − 0.591i)13-s − 1.62i·17-s + 0.397i·19-s + (−0.618 + 1.07i)23-s + (−0.392 − 0.680i)25-s + (−0.332 + 0.192i)29-s + (0.0833 + 0.0481i)31-s + 0.0468·35-s − 1.06·37-s + (0.280 + 0.161i)41-s + (−0.985 + 0.569i)43-s + (−0.694 − 1.20i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.048845528\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.048845528\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.896 - 0.517i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.232 + 0.133i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.93 + 3.34i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.23 + 2.13i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6.69iT - 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + (2.96 - 5.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.79 - 1.03i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.464 - 0.267i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.46T + 37T^{2} \) |
| 41 | \( 1 + (-1.79 - 1.03i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.46 - 3.73i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.76 + 8.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 + (3.72 - 6.45i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.69 - 8.13i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.42 + 4.86i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.92T + 73T^{2} \) |
| 79 | \( 1 + (-13.1 + 7.59i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.86 + 6.69i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6.69iT - 89T^{2} \) |
| 97 | \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.534949716594310960760681417562, −7.903418981741297323928962717991, −7.14752643468869436820801903402, −6.18300514050920211197724322617, −5.54954431235949392898506197946, −4.84084479859969412370686440129, −3.53518287124088793731948816754, −2.93668811663237612513489289489, −1.75602049487908321683763271274, −0.32209650684622025498821876561,
1.58655369161374424983736896236, 2.24767808248478715270487245374, 3.59418964514011045878615014595, 4.43652896698899507362973936811, 5.21180330576128224504803383902, 6.12422933288680268519426148491, 6.75264880319503256452476914875, 7.72456833416808650018398056109, 8.377819223051804887751825529840, 9.147587314723591259057561032660