| L(s) = 1 | − 0.381·2-s − 3.85·4-s − 3.70i·7-s + 3·8-s + 8.18i·11-s + 7.70i·13-s + 1.41i·14-s + 14.2·16-s − 11.9·17-s − 5.58·19-s − 3.12i·22-s + 28.4·23-s − 2.94i·26-s + 14.2i·28-s − 56.0i·29-s + ⋯ |
| L(s) = 1 | − 0.190·2-s − 0.963·4-s − 0.529i·7-s + 0.375·8-s + 0.743i·11-s + 0.592i·13-s + 0.101i·14-s + 0.891·16-s − 0.702·17-s − 0.293·19-s − 0.142i·22-s + 1.23·23-s − 0.113i·26-s + 0.510i·28-s − 1.93i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9913317642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9913317642\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 4T^{2} \) |
| 7 | \( 1 + 3.70iT - 49T^{2} \) |
| 11 | \( 1 - 8.18iT - 121T^{2} \) |
| 13 | \( 1 - 7.70iT - 169T^{2} \) |
| 17 | \( 1 + 11.9T + 289T^{2} \) |
| 19 | \( 1 + 5.58T + 361T^{2} \) |
| 23 | \( 1 - 28.4T + 529T^{2} \) |
| 29 | \( 1 + 56.0iT - 841T^{2} \) |
| 31 | \( 1 + 31.2T + 961T^{2} \) |
| 37 | \( 1 - 53.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 60.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 71.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 73.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 90.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 19T + 3.72e3T^{2} \) |
| 67 | \( 1 + 0.334iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 81.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 50.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 48.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 62.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 69.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 159. iT - 9.40e3T^{2} \) |
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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
−10.07501928484043637359091072363, −9.226295340921019618590110512940, −8.609164099416849388123088318543, −7.50962050638129073451947341382, −6.78356544654656688905114078559, −5.45254352102942392684205511720, −4.48064371440739818998523779257, −3.80361416226858691100775212144, −2.08857353074659077167218940236, −0.49777516592633552374562985781,
1.02369648689540047246564834341, 2.79542670605320430831441066188, 3.91430276294251798861329251723, 5.07740470677467240497964890959, 5.74561786195739008089958527295, 7.00670691320117785594866374885, 8.051933893523162688655960584654, 8.959477308888416163947221234656, 9.203089035911128756037244956505, 10.56276458359496995386436776281
