| L(s) = 1 | + (1.62 + 0.345i)2-s + (1.73 − 0.0263i)3-s + (0.700 + 0.311i)4-s + (−0.380 + 2.20i)5-s + (2.82 + 0.556i)6-s + (0.694 − 1.20i)7-s + (−1.65 − 1.20i)8-s + (2.99 − 0.0912i)9-s + (−1.38 + 3.45i)10-s + (−4.44 − 0.945i)11-s + (1.22 + 0.521i)12-s + (−0.664 + 0.141i)13-s + (1.54 − 1.71i)14-s + (−0.600 + 3.82i)15-s + (−3.30 − 3.67i)16-s + (−1.73 − 1.26i)17-s + ⋯ |
| L(s) = 1 | + (1.15 + 0.244i)2-s + (0.999 − 0.0152i)3-s + (0.350 + 0.155i)4-s + (−0.170 + 0.985i)5-s + (1.15 + 0.227i)6-s + (0.262 − 0.454i)7-s + (−0.586 − 0.426i)8-s + (0.999 − 0.0304i)9-s + (−0.436 + 1.09i)10-s + (−1.34 − 0.285i)11-s + (0.352 + 0.150i)12-s + (−0.184 + 0.0391i)13-s + (0.412 − 0.458i)14-s + (−0.155 + 0.987i)15-s + (−0.827 − 0.918i)16-s + (−0.420 − 0.305i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.42271 + 0.509504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.42271 + 0.509504i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.73 + 0.0263i)T \) |
| 5 | \( 1 + (0.380 - 2.20i)T \) |
| good | 2 | \( 1 + (-1.62 - 0.345i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (-0.694 + 1.20i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.44 + 0.945i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.664 - 0.141i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (1.73 + 1.26i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-5.56 - 4.04i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.0716 - 0.0796i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.214 + 2.04i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.675 + 6.43i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.886 - 2.72i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (5.31 - 1.12i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (0.613 - 1.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.23 - 11.7i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (1.07 - 0.781i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.69 + 0.360i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.69 - 0.784i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.48 + 14.1i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (10.9 - 7.96i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.83 + 5.63i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.938 - 8.92i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-16.5 + 7.37i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.33 - 7.18i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.13 + 10.8i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.67811232805499881378714893993, −11.52074633548990246838963232657, −10.33671590523514363910506649728, −9.521322006808196731436314757231, −7.957954839217503075566435728056, −7.33870255111197942477862852208, −6.06500306007111575927968953959, −4.74249989687992916076248844896, −3.57517927104092501115729269047, −2.66938562741370965641831640010,
2.25627862611603117238267629582, 3.47272153294568382887779356335, 4.81323310974343178653809748904, 5.32167349217003963713807358443, 7.24103486640473338727454793549, 8.426419873932933173213803403431, 8.981941375714635759788139601832, 10.24091986029774399532436238237, 11.68504488937125704562246448208, 12.48241574158675247899826866652
