L(s) = 1 | + 1.89i·2-s + (0.180 + 2.99i)3-s + 0.403·4-s − 2.43i·5-s + (−5.67 + 0.342i)6-s − 8.99·7-s + 8.35i·8-s + (−8.93 + 1.08i)9-s + 4.62·10-s + 9.24i·11-s + (0.0729 + 1.20i)12-s − 11.6·13-s − 17.0i·14-s + (7.30 − 0.440i)15-s − 14.2·16-s + 27.6i·17-s + ⋯ |
L(s) = 1 | + 0.948i·2-s + (0.0602 + 0.998i)3-s + 0.100·4-s − 0.487i·5-s + (−0.946 + 0.0571i)6-s − 1.28·7-s + 1.04i·8-s + (−0.992 + 0.120i)9-s + 0.462·10-s + 0.840i·11-s + (0.00608 + 0.100i)12-s − 0.896·13-s − 1.21i·14-s + (0.486 − 0.0293i)15-s − 0.888·16-s + 1.62i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0602i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0363917 - 1.20695i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0363917 - 1.20695i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.180 - 2.99i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 2 | \( 1 - 1.89iT - 4T^{2} \) |
| 5 | \( 1 + 2.43iT - 25T^{2} \) |
| 7 | \( 1 + 8.99T + 49T^{2} \) |
| 11 | \( 1 - 9.24iT - 121T^{2} \) |
| 13 | \( 1 + 11.6T + 169T^{2} \) |
| 17 | \( 1 - 27.6iT - 289T^{2} \) |
| 19 | \( 1 - 29.8T + 361T^{2} \) |
| 23 | \( 1 + 29.8iT - 529T^{2} \) |
| 29 | \( 1 - 16.6iT - 841T^{2} \) |
| 31 | \( 1 + 11.5T + 961T^{2} \) |
| 37 | \( 1 - 68.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 2.67iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 14.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 15.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 67.2iT - 2.80e3T^{2} \) |
| 61 | \( 1 - 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 55.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 123. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 81.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 45.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 144. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 8.00iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 41.6T + 9.40e3T^{2} \) |
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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
−12.92325625441774294692111734639, −12.11146368358208547865487027974, −10.73324584061007040187580580810, −9.810162774617688349816783597006, −8.968021778772664323351546338913, −7.77605578153751138394386658074, −6.59222630286824957988903006766, −5.56648847012034279232249527261, −4.43461389151825086163542395513, −2.81722780940183651087035167616,
0.69574672383740720267144236572, 2.69344253421740060260963411048, 3.24596228157815470525803297471, 5.68156756273905544654243701281, 6.89830974449278095367768051335, 7.45935570393606055539119695318, 9.316022235643213997207072699916, 9.936679302500684731617474087109, 11.49121365763809645365554891508, 11.66542908419742452010290051727