| L(s) = 1 | + (−2.82 − i)3-s + (7.00 + 5.65i)9-s + 14i·11-s − 33.9i·17-s − 16.9·19-s + 25·25-s + (−14.1 − 23.0i)27-s + (14 − 39.5i)33-s + 67.8i·41-s − 84.8·43-s − 49·49-s + (−33.9 + 96i)51-s + (48 + 16.9i)57-s + 82i·59-s − 118.·67-s + ⋯ |
| L(s) = 1 | + (−0.942 − 0.333i)3-s + (0.777 + 0.628i)9-s + 1.27i·11-s − 1.99i·17-s − 0.893·19-s + 25-s + (−0.523 − 0.851i)27-s + (0.424 − 1.19i)33-s + 1.65i·41-s − 1.97·43-s − 0.999·49-s + (−0.665 + 1.88i)51-s + (0.842 + 0.297i)57-s + 1.38i·59-s − 1.77·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1139276225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1139276225\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.82 + i)T \) |
| good | 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 14iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 33.9iT - 289T^{2} \) |
| 19 | \( 1 + 16.9T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 67.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 84.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 82iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 118.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 142T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 + 158iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
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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
−10.44174469627434646853694748610, −9.865067662805300368201728009483, −8.877081744942437491707895978961, −7.61332373380343448830432751761, −7.00879917454896232851109247840, −6.23181953872864823937099783579, −4.93195712800416955711357331986, −4.58995259995305007823911337123, −2.80728873216392143731892900783, −1.48952268650664630491492644701,
0.04569525081837175703058432778, 1.54805795712717115100405820466, 3.35805764807088626954579788717, 4.24717716778894650797233325681, 5.38126678357049588843816289882, 6.15659636866224041290773310214, 6.80992643578864394722428814452, 8.225621436517591061926294574118, 8.790313956913252331520393516660, 10.00888202454243871328464342123
