L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.741 − 1.28i)3-s + (−0.499 − 0.866i)4-s + (0.741 + 1.28i)6-s + 0.482·7-s + 0.999·8-s + (0.400 + 0.693i)9-s − 4.43·11-s − 1.48·12-s + (2.07 + 3.58i)13-s + (−0.241 + 0.418i)14-s + (−0.5 + 0.866i)16-s + (−3.94 + 6.84i)17-s − 0.801·18-s + (4.31 + 0.590i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.428 − 0.741i)3-s + (−0.249 − 0.433i)4-s + (0.302 + 0.524i)6-s + 0.182·7-s + 0.353·8-s + (0.133 + 0.231i)9-s − 1.33·11-s − 0.428·12-s + (0.574 + 0.994i)13-s + (−0.0645 + 0.111i)14-s + (−0.125 + 0.216i)16-s + (−0.957 + 1.65i)17-s − 0.188·18-s + (0.990 + 0.135i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16233 + 0.732144i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16233 + 0.732144i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4.31 - 0.590i)T \) |
good | 3 | \( 1 + (-0.741 + 1.28i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 0.482T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 + (-2.07 - 3.58i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.94 - 6.84i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.82 - 4.90i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.91 + 3.31i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 - 3.50T + 37T^{2} \) |
| 41 | \( 1 + (-4.44 + 7.70i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.318 - 0.551i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.13 + 3.69i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.59 - 2.76i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.812 - 1.40i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.735 + 1.27i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.59 - 11.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.41 - 2.44i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.43 - 5.95i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.576 + 0.997i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + (4.37 + 7.58i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.476 - 0.825i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.12158419986959192931696458462, −9.101485323944194487533197945108, −8.270557306341265599896474512643, −7.78123221747833576407615706770, −6.96436591057964004487705529773, −6.09478479609371042436011636936, −5.11062816817042781290695704041, −4.01438021442168995116662723966, −2.46196984709624954145449595615, −1.41922755625064251589519967907,
0.74665402422251132205667478924, 2.71873771090678611083702434964, 3.14335981894938167360761445461, 4.57668300721514446930735031087, 5.11358553358361030698461036763, 6.57649345071213982663330957464, 7.69669008371599175685334717375, 8.361980301158609172963175233556, 9.302846953267250159650088315700, 9.806333494128020400173536541732