| L(s) = 1 | − 2-s + (−1.65 + 0.500i)3-s + 4-s − 0.650i·5-s + (1.65 − 0.500i)6-s − 1.64i·7-s − 8-s + (2.49 − 1.65i)9-s + 0.650i·10-s + (1.52 − 2.94i)11-s + (−1.65 + 0.500i)12-s + 3.18i·13-s + 1.64i·14-s + (0.325 + 1.07i)15-s + 16-s + 17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.957 + 0.288i)3-s + 0.5·4-s − 0.290i·5-s + (0.676 − 0.204i)6-s − 0.621i·7-s − 0.353·8-s + (0.832 − 0.553i)9-s + 0.205i·10-s + (0.459 − 0.888i)11-s + (−0.478 + 0.144i)12-s + 0.882i·13-s + 0.439i·14-s + (0.0840 + 0.278i)15-s + 0.250·16-s + 0.242·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6138789511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6138789511\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.65 - 0.500i)T \) |
| 11 | \( 1 + (-1.52 + 2.94i)T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 0.650iT - 5T^{2} \) |
| 7 | \( 1 + 1.64iT - 7T^{2} \) |
| 13 | \( 1 - 3.18iT - 13T^{2} \) |
| 19 | \( 1 - 0.117iT - 19T^{2} \) |
| 23 | \( 1 + 4.37iT - 23T^{2} \) |
| 29 | \( 1 + 8.10T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 - 0.687T + 37T^{2} \) |
| 41 | \( 1 + 3.22T + 41T^{2} \) |
| 43 | \( 1 - 6.63iT - 43T^{2} \) |
| 47 | \( 1 + 0.193iT - 47T^{2} \) |
| 53 | \( 1 + 5.76iT - 53T^{2} \) |
| 59 | \( 1 + 10.9iT - 59T^{2} \) |
| 61 | \( 1 + 1.38iT - 61T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 + 13.7iT - 71T^{2} \) |
| 73 | \( 1 + 1.19iT - 73T^{2} \) |
| 79 | \( 1 + 16.8iT - 79T^{2} \) |
| 83 | \( 1 + 5.71T + 83T^{2} \) |
| 89 | \( 1 + 4.71iT - 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−9.546143086207725626275237053768, −9.008627326864906539644203888775, −7.992594996267484341868211905498, −6.95883185139510117862260680573, −6.40906522025583009528792707443, −5.42518503126817881155965930039, −4.39337774686260295331964865854, −3.44967423993082458354080696245, −1.62551620705121917169550933720, −0.43048540629650678120028748603,
1.29367259577811743967759150956, 2.46345747613774980658373201147, 3.89836489829540671879293327993, 5.31458593200569044348338747206, 5.78604794049958369643910384718, 6.99903728528008078536295377285, 7.33157207238276845221396485652, 8.407855672505349848507377402227, 9.403611211243300841584823077363, 10.04811633869189716315076280818
