| L(s) = 1 | − 5.19·7-s + 18.9i·11-s + 13i·13-s + 32.8i·17-s + 15.5i·19-s − 18.9·23-s − 32.8·29-s − 32.9i·31-s + 46i·37-s − 32.8·41-s + 12.1·43-s + 37.9·47-s − 22·49-s − 32.8i·53-s + 37.9i·59-s + ⋯ |
| L(s) = 1 | − 0.742·7-s + 1.72i·11-s + i·13-s + 1.93i·17-s + 0.820i·19-s − 0.824·23-s − 1.13·29-s − 1.06i·31-s + 1.24i·37-s − 0.801·41-s + 0.281·43-s + 0.807·47-s − 0.448·49-s − 0.620i·53-s + 0.643i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6850170202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6850170202\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5.19T + 49T^{2} \) |
| 11 | \( 1 - 18.9iT - 121T^{2} \) |
| 13 | \( 1 - 13iT - 169T^{2} \) |
| 17 | \( 1 - 32.8iT - 289T^{2} \) |
| 19 | \( 1 - 15.5iT - 361T^{2} \) |
| 23 | \( 1 + 18.9T + 529T^{2} \) |
| 29 | \( 1 + 32.8T + 841T^{2} \) |
| 31 | \( 1 + 32.9iT - 961T^{2} \) |
| 37 | \( 1 - 46iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 12.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 37.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 32.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 37.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 59T + 3.72e3T^{2} \) |
| 67 | \( 1 + 39.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 94.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 26iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 138. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 113.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 131.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 23iT - 9.40e3T^{2} \) |
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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
−8.912788967432165706963486666003, −7.946066395040994955315370149543, −7.41330261403181512732157574137, −6.39499891833677606664308533837, −6.14392747386974241570581352154, −4.94404302436543642936932774721, −4.07646373335524502012731593859, −3.61210676992557779230262780726, −2.11308683764885202657467652720, −1.68796154747533462173071459824,
0.17935430017882780024774587622, 0.826678289806531481913426273358, 2.50522273915893187565718577735, 3.15256979994435588245320872868, 3.83930661336546466781737012627, 5.16462641854984852008906155972, 5.58414373825667944228414784202, 6.45317679444823698771053211432, 7.20035668109005128860364284227, 7.924483996111118843243522572369
