| L(s) = 1 | − 2i·2-s − 4·4-s + 4i·7-s + 8i·8-s + 8·14-s + 16·16-s − 44i·23-s − 16i·28-s − 22·29-s − 32i·32-s − 62·41-s − 76i·43-s − 88·46-s − 4i·47-s + 33·49-s + ⋯ |
| L(s) = 1 | − i·2-s − 4-s + 0.571i·7-s + i·8-s + 0.571·14-s + 16-s − 1.91i·23-s − 0.571i·28-s − 0.758·29-s − i·32-s − 1.51·41-s − 1.76i·43-s − 1.91·46-s − 0.0851i·47-s + 0.673·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7948608707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7948608707\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 44iT - 529T^{2} \) |
| 29 | \( 1 + 22T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 62T + 1.68e3T^{2} \) |
| 43 | \( 1 + 76iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 58T + 3.72e3T^{2} \) |
| 67 | \( 1 + 116iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 76iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 142T + 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−9.577876250556927170591658690110, −8.769657172680561004186087582568, −8.191167175440060067347233250276, −6.91893708378497632174186433153, −5.77483167076938673251207450493, −4.90066039441487419300059224535, −3.90978388260583973806455883730, −2.80536450053952494591029527338, −1.84764295490495388321207393378, −0.27367963529671785378984330852,
1.34114430161747284119835119523, 3.29678485127138449705878856317, 4.21309215128560958340864887298, 5.23283913121832275437019482491, 6.03643660034004665859462621001, 7.07142534181749433030571125740, 7.62029537750107254286895566345, 8.507174535346738233606356686592, 9.448083517539989248705096480525, 10.02774104821915182413621120170
