L(s) = 1 | + 5.38i·3-s + 4·5-s + 7-s − 19.9·9-s + 14·11-s − 16.1i·13-s + 21.5i·15-s + 23·17-s + (10 + 16.1i)19-s + 5.38i·21-s + 23-s − 9·25-s − 59.2i·27-s + 48.4i·29-s + 32.3i·31-s + ⋯ |
L(s) = 1 | + 1.79i·3-s + 0.800·5-s + 0.142·7-s − 2.22·9-s + 1.27·11-s − 1.24i·13-s + 1.43i·15-s + 1.35·17-s + (0.526 + 0.850i)19-s + 0.256i·21-s + 0.0434·23-s − 0.359·25-s − 2.19i·27-s + 1.67i·29-s + 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.470340094\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.470340094\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-10 - 16.1i)T \) |
good | 3 | \( 1 - 5.38iT - 9T^{2} \) |
| 5 | \( 1 - 4T + 25T^{2} \) |
| 7 | \( 1 - T + 49T^{2} \) |
| 11 | \( 1 - 14T + 121T^{2} \) |
| 13 | \( 1 + 16.1iT - 169T^{2} \) |
| 17 | \( 1 - 23T + 289T^{2} \) |
| 23 | \( 1 - T + 529T^{2} \) |
| 29 | \( 1 - 48.4iT - 841T^{2} \) |
| 31 | \( 1 - 32.3iT - 961T^{2} \) |
| 37 | \( 1 + 32.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 68T + 1.84e3T^{2} \) |
| 47 | \( 1 + 26T + 2.20e3T^{2} \) |
| 53 | \( 1 - 80.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 16.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 40T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 32.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 96.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 32T + 6.88e3T^{2} \) |
| 89 | \( 1 + 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 96.9iT - 9.40e3T^{2} \) |
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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
−9.849767427588763911175565021905, −9.260830292487217014082670146261, −8.522736283913833552330693976924, −7.47590301381111272033370804583, −5.98887171304563702153917503313, −5.57991657693264588891736896762, −4.71885420762624143477846122402, −3.61064707848301841295365696928, −3.08577152874536546932261570591, −1.30987901373179710315103427509,
0.818038274152329333782654862143, 1.70870053072043805899752136022, 2.48785799172814581395880074537, 3.90215098099036191626260203998, 5.35349135037206143523153771266, 6.24332934296524810959101456604, 6.65462470832823713973030884777, 7.54129496404459309833062785879, 8.249045386689032405343335985171, 9.322302678789353060981181437082