| L(s) = 1 | + (0.0713 − 0.997i)2-s + (−2.03 + 0.442i)3-s + (−0.989 − 0.142i)4-s + (2.15 + 0.596i)5-s + (0.296 + 2.05i)6-s + (−3.45 + 1.88i)7-s + (−0.212 + 0.977i)8-s + (1.20 − 0.552i)9-s + (0.748 − 2.10i)10-s + (3.04 + 2.63i)11-s + (2.07 − 0.148i)12-s + (−1.68 + 3.08i)13-s + (1.63 + 3.58i)14-s + (−4.64 − 0.260i)15-s + (0.959 + 0.281i)16-s + (4.11 + 3.08i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (−1.17 + 0.255i)3-s + (−0.494 − 0.0711i)4-s + (0.963 + 0.266i)5-s + (0.120 + 0.840i)6-s + (−1.30 + 0.713i)7-s + (−0.0751 + 0.345i)8-s + (0.403 − 0.184i)9-s + (0.236 − 0.666i)10-s + (0.916 + 0.794i)11-s + (0.599 − 0.0428i)12-s + (−0.467 + 0.856i)13-s + (0.437 + 0.957i)14-s + (−1.19 − 0.0671i)15-s + (0.239 + 0.0704i)16-s + (0.998 + 0.747i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.599538 + 0.350142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.599538 + 0.350142i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 5 | \( 1 + (-2.15 - 0.596i)T \) |
| 23 | \( 1 + (4.12 - 2.44i)T \) |
| good | 3 | \( 1 + (2.03 - 0.442i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (3.45 - 1.88i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-3.04 - 2.63i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.68 - 3.08i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-4.11 - 3.08i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.239 + 1.66i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (4.67 - 0.671i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (4.96 - 3.19i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (0.514 + 0.192i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-2.91 + 6.37i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.214 - 0.984i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (8.34 + 8.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.06 + 1.94i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-3.42 - 11.6i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (1.10 + 1.72i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-5.64 - 0.403i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-8.02 - 9.26i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.73 - 3.65i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-13.5 + 3.98i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-4.13 + 11.0i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-5.99 - 3.84i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.678 - 1.81i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−12.24959244643675743538578248041, −11.53574346403844183431913097860, −10.35647711261361108760004702035, −9.714263173340062290781197856817, −9.068033308562423850786386624644, −6.91352475462607851223569627687, −6.04162476454889363824507235698, −5.23666822224097336661105511273, −3.67181405374703653858571767547, −2.01175997901873379009750941162,
0.65091536933177783831979574352, 3.45591684936246161660791690030, 5.17670148547653319030575741295, 6.08003644122888924241508243351, 6.52518061707099556763949655739, 7.79642396145919695367583452930, 9.381385623167450582396727401240, 9.927593315236557670774807282995, 11.05200443235686294362473930249, 12.33603077539884275787687176699
