L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 5·7-s − 0.999·8-s + (0.499 − 0.866i)10-s − 2·11-s + (0.5 − 0.866i)13-s + (−2.5 − 4.33i)14-s + (−0.5 − 0.866i)16-s + (3 + 5.19i)17-s + (4 − 1.73i)19-s + 0.999·20-s + (−1 − 1.73i)22-s + (2 − 3.46i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 1.88·7-s − 0.353·8-s + (0.158 − 0.273i)10-s − 0.603·11-s + (0.138 − 0.240i)13-s + (−0.668 − 1.15i)14-s + (−0.125 − 0.216i)16-s + (0.727 + 1.26i)17-s + (0.917 − 0.397i)19-s + 0.223·20-s + (−0.213 − 0.369i)22-s + (0.417 − 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.326329925\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.326329925\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 7 | \( 1 + 5T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.5 - 7.79i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4 + 6.92i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14T + 83T^{2} \) |
| 89 | \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.429089659863224028451758567767, −8.359209250366566383482817984330, −7.82502471249014003924523544978, −6.74272068099818504588339778495, −6.21693854887701469238238470277, −5.43953668065188303888367186633, −4.39116918008156980915547788361, −3.43288489374496131619734322575, −2.74891271415940259379588602673, −0.62481025448064491420445657658,
0.876544411367562825664489324460, 2.80092832660182864791353993374, 3.06376441912086079563457732366, 4.03885958995569220047125673723, 5.27243818145011834497810027987, 5.95995767318720742295071986449, 6.95374338711809485863468420884, 7.47212817433535690845238684089, 8.810447015138742236990989367848, 9.546731506986559209725891399938