L(s) = 1 | − 2-s + (−1.02 + 1.39i)3-s + 4-s + (0.5 + 0.866i)5-s + (1.02 − 1.39i)6-s + (−2.52 + 0.795i)7-s − 8-s + (−0.918 − 2.85i)9-s + (−0.5 − 0.866i)10-s + (2.96 − 5.14i)11-s + (−1.02 + 1.39i)12-s + (2.58 − 4.47i)13-s + (2.52 − 0.795i)14-s + (−1.72 − 0.183i)15-s + 16-s + (3.72 + 6.46i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.588 + 0.808i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (0.416 − 0.571i)6-s + (−0.953 + 0.300i)7-s − 0.353·8-s + (−0.306 − 0.951i)9-s + (−0.158 − 0.273i)10-s + (0.894 − 1.54i)11-s + (−0.294 + 0.404i)12-s + (0.717 − 1.24i)13-s + (0.674 − 0.212i)14-s + (−0.444 − 0.0473i)15-s + 0.250·16-s + (0.904 + 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.847439 + 0.141512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.847439 + 0.141512i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.02 - 1.39i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.52 - 0.795i)T \) |
good | 11 | \( 1 + (-2.96 + 5.14i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.58 + 4.47i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.72 - 6.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.78 + 3.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.367 + 0.635i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.46 - 4.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.93T + 31T^{2} \) |
| 37 | \( 1 + (-0.0338 + 0.0585i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.03 - 6.99i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.84 + 6.65i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + (-2.41 - 4.17i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 2.55T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 6.07T + 67T^{2} \) |
| 71 | \( 1 - 5.08T + 71T^{2} \) |
| 73 | \( 1 + (-0.250 - 0.433i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 7.73T + 79T^{2} \) |
| 83 | \( 1 + (2.63 + 4.57i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.43 + 4.21i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.484 + 0.839i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.53770899882903709329317453315, −9.945267235189693428217614050224, −8.900348728438647732795867531882, −8.432425692866661983710112189112, −6.89968503463925803058855797134, −5.95159119787599500890298598759, −5.67826156047646465413662526087, −3.62195477544153621109262493417, −3.18495903759112065820338445337, −0.859189649824870529058602089556,
1.04524203634278503185634118968, 2.19154741720087814963968006267, 3.91508962130826350158639148940, 5.26999444067432165756900713395, 6.41112919882615429256600911778, 6.98914356804995285233229347878, 7.68189225065363916949281361868, 9.033359393868298629387271452564, 9.622029946905517233135596406710, 10.34201259818894197623861034090