L(s) = 1 | + 2-s + 4-s + (−1.5 − 2.59i)5-s + (2.5 − 0.866i)7-s + 8-s + (−1.5 − 2.59i)10-s + (1.5 − 2.59i)11-s + (−1 + 1.73i)13-s + (2.5 − 0.866i)14-s + 16-s + (−3 − 5.19i)17-s + (−1 + 1.73i)19-s + (−1.5 − 2.59i)20-s + (1.5 − 2.59i)22-s + (3 + 5.19i)23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.670 − 1.16i)5-s + (0.944 − 0.327i)7-s + 0.353·8-s + (−0.474 − 0.821i)10-s + (0.452 − 0.783i)11-s + (−0.277 + 0.480i)13-s + (0.668 − 0.231i)14-s + 0.250·16-s + (−0.727 − 1.26i)17-s + (−0.229 + 0.397i)19-s + (−0.335 − 0.580i)20-s + (0.319 − 0.553i)22-s + (0.625 + 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0477 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0477 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.304879987\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.304879987\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.283595896992974442075476352530, −8.929852928436489647698857220375, −7.70767436708935419406150857337, −7.38724254031940482023510481459, −5.98214044708989093621057103376, −5.15859272159296445261864867054, −4.36619023701247332718681478685, −3.76474517141537055112774050366, −2.16680933705359737953008220227, −0.818474004284224395541938613026,
1.84675008184175019533426816968, 2.86528573219665802794138618280, 3.95060500408206712452364208494, 4.69444558017019251894814330617, 5.74691084702890531355335921589, 6.82444816796319739797445701865, 7.26367851897535541245973555076, 8.225506060863189531468402953674, 9.093063828339997281485906691852, 10.48540264147949102059502109763