| L(s) = 1 | + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + 6.92i·7-s + 7.99·8-s − 6.92i·11-s − 2·13-s + (11.9 − 6.92i)14-s + (−8 − 13.8i)16-s + 10·17-s + 20.7i·19-s + (−11.9 + 6.92i)22-s − 27.7i·23-s + (2 + 3.46i)26-s + (−23.9 − 13.8i)28-s + 26·29-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.989i·7-s + 0.999·8-s − 0.629i·11-s − 0.153·13-s + (0.857 − 0.494i)14-s + (−0.5 − 0.866i)16-s + 0.588·17-s + 1.09i·19-s + (−0.545 + 0.314i)22-s − 1.20i·23-s + (0.0769 + 0.133i)26-s + (−0.857 − 0.494i)28-s + 0.896·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9168284647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9168284647\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6.92iT - 49T^{2} \) |
| 11 | \( 1 + 6.92iT - 121T^{2} \) |
| 13 | \( 1 + 2T + 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 - 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 27.7iT - 529T^{2} \) |
| 29 | \( 1 - 26T + 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 26T + 1.36e3T^{2} \) |
| 41 | \( 1 + 58T + 1.68e3T^{2} \) |
| 43 | \( 1 - 48.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 74T + 2.80e3T^{2} \) |
| 59 | \( 1 - 90.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.92iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 82T + 7.92e3T^{2} \) |
| 97 | \( 1 + 2T + 9.40e3T^{2} \) |
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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
−10.10121736211881363732489087483, −9.276319658595028567422831167300, −8.426193407961653379349674987471, −7.990747665594427114415226153817, −6.65475508742747312508217369889, −5.62162409105290670025508667788, −4.57737962263966161327203843505, −3.36038304632522444304701878812, −2.53214787722504622801103474391, −1.25159932223937017415921956428,
0.39056070180863128086165661181, 1.72820330924971449169097707782, 3.53769149839324454120994755918, 4.67716430119025593109464936858, 5.39107564862707053524326507204, 6.68965475963216073109836709197, 7.15904640976172747630210728069, 7.951735487941129261012602524697, 8.869219056415521037590423395857, 9.801363894747231256086052369534
