| L(s) = 1 | + (0.256 + 0.0545i)3-s + (0.121 − 1.15i)5-s + (−1.12 − 2.39i)7-s + (−2.67 − 1.19i)9-s + (2.47 − 2.20i)11-s + (3.85 − 2.80i)13-s + (0.0942 − 0.289i)15-s + (1.06 − 0.474i)17-s + (−3.68 + 4.09i)19-s + (−0.159 − 0.675i)21-s + (0.306 + 0.530i)23-s + (3.56 + 0.758i)25-s + (−1.25 − 0.914i)27-s + (−0.689 + 2.12i)29-s + (−0.445 − 4.23i)31-s + ⋯ |
| L(s) = 1 | + (0.148 + 0.0314i)3-s + (0.0543 − 0.516i)5-s + (−0.426 − 0.904i)7-s + (−0.892 − 0.397i)9-s + (0.746 − 0.665i)11-s + (1.07 − 0.777i)13-s + (0.0243 − 0.0748i)15-s + (0.258 − 0.115i)17-s + (−0.845 + 0.938i)19-s + (−0.0347 − 0.147i)21-s + (0.0639 + 0.110i)23-s + (0.713 + 0.151i)25-s + (−0.242 − 0.176i)27-s + (−0.128 + 0.394i)29-s + (−0.0799 − 0.761i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02585 - 0.718489i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02585 - 0.718489i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.12 + 2.39i)T \) |
| 11 | \( 1 + (-2.47 + 2.20i)T \) |
| good | 3 | \( 1 + (-0.256 - 0.0545i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.121 + 1.15i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (-3.85 + 2.80i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.06 + 0.474i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (3.68 - 4.09i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.306 - 0.530i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.689 - 2.12i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.445 + 4.23i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-10.7 + 2.28i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (1.76 + 5.43i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 + (8.14 - 9.04i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-1.15 - 10.9i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-3.43 - 3.81i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.897 + 8.53i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (3.54 - 6.14i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.17 - 5.21i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.65 - 1.84i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-7.36 - 3.27i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (11.5 + 8.35i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-4.69 - 8.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.44 + 4.68i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.37943517610530132498532925231, −10.69811634346529779319412927411, −9.538951582956945789036629500793, −8.670602290138157630999497630728, −7.88242534259752755142236994287, −6.42866734880344158858417303465, −5.73004180010087609868307595088, −4.09724543280317080127090378315, −3.21023167042572334810603441078, −0.964146125966598977514517716260,
2.14851588900893797495254432851, 3.34936020193368494414067672665, 4.82324831068166884932527266212, 6.21433439672235401554451307874, 6.75504683340479367364779701043, 8.332595179486842026999512494289, 8.949605912792520924223720513134, 9.932453185392601013646514420818, 11.18655372033503100234246088040, 11.65581278290466887694123002710
