L(s) = 1 | + (1.47 + 0.900i)3-s + (1.26 − 2.18i)5-s + (−2.63 + 0.275i)7-s + (1.37 + 2.66i)9-s + (−2.85 − 4.94i)11-s + (−2.45 − 4.24i)13-s + (3.83 − 2.09i)15-s + (2.49 − 4.32i)17-s + (0.00383 + 0.00664i)19-s + (−4.14 − 1.96i)21-s + (0.333 − 0.578i)23-s + (−0.682 − 1.18i)25-s + (−0.355 + 5.18i)27-s + (3.85 − 6.66i)29-s + 7.76·31-s + ⋯ |
L(s) = 1 | + (0.854 + 0.519i)3-s + (0.564 − 0.977i)5-s + (−0.994 + 0.104i)7-s + (0.459 + 0.887i)9-s + (−0.861 − 1.49i)11-s + (−0.680 − 1.17i)13-s + (0.989 − 0.541i)15-s + (0.605 − 1.04i)17-s + (0.000880 + 0.00152i)19-s + (−0.903 − 0.427i)21-s + (0.0696 − 0.120i)23-s + (−0.136 − 0.236i)25-s + (−0.0685 + 0.997i)27-s + (0.715 − 1.23i)29-s + 1.39·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.819478113\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.819478113\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.47 - 0.900i)T \) |
| 7 | \( 1 + (2.63 - 0.275i)T \) |
good | 5 | \( 1 + (-1.26 + 2.18i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.85 + 4.94i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.45 + 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.49 + 4.32i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.00383 - 0.00664i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.333 + 0.578i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.85 + 6.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7.76T + 31T^{2} \) |
| 37 | \( 1 + (3.19 + 5.53i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.21 - 9.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.42 - 7.67i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 2.16T + 47T^{2} \) |
| 53 | \( 1 + (3.69 - 6.40i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 0.523T + 59T^{2} \) |
| 61 | \( 1 + 8.99T + 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 - 5.68T + 71T^{2} \) |
| 73 | \( 1 + (1.52 - 2.63i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 6.16T + 79T^{2} \) |
| 83 | \( 1 + (-0.258 + 0.448i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.19 - 2.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.32 + 7.49i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.821886101032511580110334451587, −9.047653821987516961366034912847, −8.220790450181162072827985393425, −7.66596362966633564743173053522, −6.16789430166659473613679886726, −5.38809840157461094694048177037, −4.61073075394130551465039881897, −3.08003783892954423458897662992, −2.77219009082083983298259975953, −0.72767643377699530884189920819,
1.86765428486627059292523297513, 2.63768731515872550789968283408, 3.58250649730876883472754207675, 4.80473231696701664757236594296, 6.26323161883886560122497337275, 6.88096386776365874254294950786, 7.36049691240422884576549094551, 8.468683272840240685961411923809, 9.464696751375693622241353299358, 10.10038380282397000756986553896