| L(s) = 1 | + (0.797 − 0.885i)2-s + (−1.56 + 0.733i)3-s + (0.0604 + 0.575i)4-s + (1.30 − 1.81i)5-s + (−0.602 + 1.97i)6-s + (0.157 + 0.272i)7-s + (2.48 + 1.80i)8-s + (1.92 − 2.30i)9-s + (−0.566 − 2.60i)10-s + (1.93 − 2.14i)11-s + (−0.516 − 0.858i)12-s + (4.36 + 4.85i)13-s + (0.367 + 0.0781i)14-s + (−0.719 + 3.80i)15-s + (2.45 − 0.521i)16-s + (0.794 + 0.577i)17-s + ⋯ |
| L(s) = 1 | + (0.564 − 0.626i)2-s + (−0.906 + 0.423i)3-s + (0.0302 + 0.287i)4-s + (0.584 − 0.811i)5-s + (−0.245 + 0.806i)6-s + (0.0595 + 0.103i)7-s + (0.879 + 0.638i)8-s + (0.641 − 0.766i)9-s + (−0.179 − 0.823i)10-s + (0.583 − 0.647i)11-s + (−0.149 − 0.247i)12-s + (1.21 + 1.34i)13-s + (0.0982 + 0.0208i)14-s + (−0.185 + 0.982i)15-s + (0.613 − 0.130i)16-s + (0.192 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45054 - 0.253220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45054 - 0.253220i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.56 - 0.733i)T \) |
| 5 | \( 1 + (-1.30 + 1.81i)T \) |
| good | 2 | \( 1 + (-0.797 + 0.885i)T + (-0.209 - 1.98i)T^{2} \) |
| 7 | \( 1 + (-0.157 - 0.272i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 2.14i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-4.36 - 4.85i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.794 - 0.577i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (5.88 + 4.27i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.28 - 0.272i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (3.94 + 1.75i)T + (19.4 + 21.5i)T^{2} \) |
| 31 | \( 1 + (7.91 - 3.52i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.00738 + 0.0227i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.06 - 3.40i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (1.34 + 2.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.76 + 2.12i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (3.44 - 2.50i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.46 - 3.84i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (4.95 - 5.50i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (2.27 - 1.01i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (-9.63 + 7.00i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.283 + 0.872i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (10.3 + 4.62i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.03 + 9.88i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-3.78 - 11.6i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (6.75 + 3.00i)T + (64.9 + 72.0i)T^{2} \) |
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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
−12.10662568912641666239290045990, −11.30746778569059809339962997235, −10.73669845571165229512577886658, −9.239921532746259974039793312548, −8.616595112729179873715562725994, −6.78500717750421154326544100337, −5.76640892117427536103499700401, −4.58484336433911971142553325720, −3.77790065659370990435504570904, −1.67777899600964975604989546505,
1.65064791605194197304253738365, 3.93607526901711181379964232576, 5.44630874779649818838713589039, 6.07046247639122664383866718575, 6.86075133844797830731214218668, 7.85462267391715482288484715704, 9.679735711954971087205908912211, 10.66475165602129073135317847377, 11.04747751980182670943521411836, 12.68805904856569402679844836819
