| L(s) = 1 | + (0.367 + 0.0781i)2-s + (0.776 − 1.54i)3-s + (−1.69 − 0.755i)4-s + (1.98 + 1.02i)5-s + (0.406 − 0.508i)6-s + (1.17 − 2.04i)7-s + (−1.17 − 0.852i)8-s + (−1.79 − 2.40i)9-s + (0.650 + 0.533i)10-s + (−0.366 − 0.0779i)11-s + (−2.48 + 2.04i)12-s + (−0.347 + 0.0738i)13-s + (0.593 − 0.659i)14-s + (3.13 − 2.27i)15-s + (2.12 + 2.35i)16-s + (3.45 + 2.51i)17-s + ⋯ |
| L(s) = 1 | + (0.260 + 0.0552i)2-s + (0.448 − 0.893i)3-s + (−0.848 − 0.377i)4-s + (0.888 + 0.459i)5-s + (0.166 − 0.207i)6-s + (0.445 − 0.772i)7-s + (−0.415 − 0.301i)8-s + (−0.597 − 0.801i)9-s + (0.205 + 0.168i)10-s + (−0.110 − 0.0235i)11-s + (−0.718 + 0.589i)12-s + (−0.0963 + 0.0204i)13-s + (0.158 − 0.176i)14-s + (0.809 − 0.587i)15-s + (0.530 + 0.589i)16-s + (0.838 + 0.609i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25544 - 0.822189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25544 - 0.822189i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.776 + 1.54i)T \) |
| 5 | \( 1 + (-1.98 - 1.02i)T \) |
| good | 2 | \( 1 + (-0.367 - 0.0781i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (-1.17 + 2.04i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.366 + 0.0779i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.347 - 0.0738i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.45 - 2.51i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.26 + 2.37i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.07 + 3.41i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.0385 - 0.366i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.614 - 5.84i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (3.03 - 9.33i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-8.44 + 1.79i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (3.86 - 6.69i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.0700 - 0.666i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (8.98 - 6.52i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-11.3 + 2.41i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (6.33 + 1.34i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.33 + 12.6i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (1.67 - 1.21i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.94 - 5.99i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.24 + 11.8i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (8.03 - 3.57i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.603 + 1.85i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.17 + 11.1i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.54067076784943903131246965953, −10.97094419340377131748410911518, −10.11445360366521673127388308270, −9.068316402726762233065101316328, −8.121784079931805320388268547478, −6.88632166699436547564759107645, −5.99422385568064221417356980363, −4.70064648047029869004798101530, −3.12500712037267605403446513401, −1.36454003263006423730485127201,
2.44387237263487055907986965014, 3.89808190769867281275385364965, 5.13594582010813456197891310255, 5.64219445669042855996959148460, 7.82528183747506749220094618559, 8.783346783635612871033754157779, 9.362022777328834976979528719620, 10.20757772567257532461454967050, 11.53675163677798709131931569403, 12.60331101694851739267875479655
