| L(s) = 1 | + (0.834 − 1.14i)2-s + (1.36 + 1.06i)3-s + (−0.605 − 1.90i)4-s + (−2.87 − 3.27i)5-s + (2.35 − 0.672i)6-s + (−1.46 − 0.192i)7-s + (−2.68 − 0.899i)8-s + (0.737 + 2.90i)9-s + (−6.13 + 0.544i)10-s + (0.246 + 0.499i)11-s + (1.19 − 3.24i)12-s + (−0.289 − 0.851i)13-s + (−1.43 + 1.50i)14-s + (−0.442 − 7.53i)15-s + (−3.26 + 2.30i)16-s + (−3.35 − 3.35i)17-s + ⋯ |
| L(s) = 1 | + (0.590 − 0.807i)2-s + (0.789 + 0.614i)3-s + (−0.302 − 0.952i)4-s + (−1.28 − 1.46i)5-s + (0.961 − 0.274i)6-s + (−0.552 − 0.0726i)7-s + (−0.948 − 0.318i)8-s + (0.245 + 0.969i)9-s + (−1.94 + 0.172i)10-s + (0.0742 + 0.150i)11-s + (0.346 − 0.938i)12-s + (−0.0801 − 0.236i)13-s + (−0.384 + 0.402i)14-s + (−0.114 − 1.94i)15-s + (−0.816 + 0.577i)16-s + (−0.812 − 0.812i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.268403 - 1.41306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.268403 - 1.41306i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.834 + 1.14i)T \) |
| 3 | \( 1 + (-1.36 - 1.06i)T \) |
| good | 5 | \( 1 + (2.87 + 3.27i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (1.46 + 0.192i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-0.246 - 0.499i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (0.289 + 0.851i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (3.35 + 3.35i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.884 + 4.44i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.887 + 6.73i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-7.68 + 0.503i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-3.00 + 1.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.35 - 6.82i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.158 + 1.20i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-3.79 + 1.87i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-1.24 + 0.334i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.29 + 4.93i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (5.51 + 6.28i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.298 + 4.55i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-12.5 - 6.21i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (1.49 + 3.60i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-0.463 + 1.11i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (1.87 + 6.98i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-3.79 - 3.32i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-11.6 + 4.84i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-16.6 - 9.59i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.34200503159123661431966396486, −9.398967447840550452052912094137, −8.807643939187331886489143441946, −8.040546610140907148657239425021, −6.68866841139831709457001582011, −4.82168277798208447800504006070, −4.72360354730826583400035883554, −3.64693714594679514880006400276, −2.57702444628973206246278009346, −0.60675587136246611421520829819,
2.64363097081133961643395106927, 3.51913227507554169693265619684, 4.14330229781543437452108602994, 6.12440859687750872209110101829, 6.66448781390543211522640365556, 7.53210708827672192671550859201, 8.021919243361491782383525293974, 8.994123862166494319126678425742, 10.21968747738849389760869701045, 11.39813456258394299977992522070
