L(s) = 1 | − 3.54·3-s − 5.75·5-s − 7.46i·7-s + 3.55·9-s + 9.81·11-s − 24.4·13-s + 20.4·15-s − 31.6i·17-s − 10.6·19-s + 26.4i·21-s + 31.8i·23-s + 8.16·25-s + 19.2·27-s + (16.2 + 23.9i)29-s + 43.2·31-s + ⋯ |
L(s) = 1 | − 1.18·3-s − 1.15·5-s − 1.06i·7-s + 0.394·9-s + 0.892·11-s − 1.88·13-s + 1.36·15-s − 1.86i·17-s − 0.562·19-s + 1.25i·21-s + 1.38i·23-s + 0.326·25-s + 0.714·27-s + (0.561 + 0.827i)29-s + 1.39·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3806265127\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3806265127\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + (-16.2 - 23.9i)T \) |
good | 3 | \( 1 + 3.54T + 9T^{2} \) |
| 5 | \( 1 + 5.75T + 25T^{2} \) |
| 7 | \( 1 + 7.46iT - 49T^{2} \) |
| 11 | \( 1 - 9.81T + 121T^{2} \) |
| 13 | \( 1 + 24.4T + 169T^{2} \) |
| 17 | \( 1 + 31.6iT - 289T^{2} \) |
| 19 | \( 1 + 10.6T + 361T^{2} \) |
| 23 | \( 1 - 31.8iT - 529T^{2} \) |
| 31 | \( 1 - 43.2T + 961T^{2} \) |
| 37 | \( 1 - 31.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 58.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 0.780T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 6.29T + 2.80e3T^{2} \) |
| 59 | \( 1 + 79.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 97.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 61.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 72.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 65.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 152.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 13.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 44.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 26.8iT - 9.40e3T^{2} \) |
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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
−11.31601055345491366661248824426, −10.19941966739331172497575355443, −9.458504133826006555674324002639, −7.964957712183780052117070154734, −7.17854560189193665204960194186, −6.57709244385942404552143443107, −4.96834720508582417032144504366, −4.53377319890528997869277400217, −3.13624471496315553461419209158, −0.828489051085108487486403810277,
0.26664419552179774332153009212, 2.37008638733329363992891206703, 4.05959608390791766789301137617, 4.89280187771612941276616378732, 6.05235907248709838862836570845, 6.71975528481630778607784228085, 8.010449685762454877295557830781, 8.722257407704124537193636467712, 9.992135859909670308924676476475, 10.85021329111716239842722280951