L(s) = 1 | + (1 + i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s − 2.44·6-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s − 2.99i·9-s + 4.18·11-s + (−2.44 − 2.44i)12-s + (12.9 − 12.9i)13-s + 3.74i·14-s − 4·16-s + (−23.0 − 23.0i)17-s + (2.99 − 2.99i)18-s − 32.6i·19-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s − 0.408·6-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + 0.380·11-s + (−0.204 − 0.204i)12-s + (0.999 − 0.999i)13-s + 0.267i·14-s − 0.250·16-s + (−1.35 − 1.35i)17-s + (0.166 − 0.166i)18-s − 1.71i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.821341977\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.821341977\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.87 - 1.87i)T \) |
good | 11 | \( 1 - 4.18T + 121T^{2} \) |
| 13 | \( 1 + (-12.9 + 12.9i)T - 169iT^{2} \) |
| 17 | \( 1 + (23.0 + 23.0i)T + 289iT^{2} \) |
| 19 | \( 1 + 32.6iT - 361T^{2} \) |
| 23 | \( 1 + (13.5 - 13.5i)T - 529iT^{2} \) |
| 29 | \( 1 + 23.2iT - 841T^{2} \) |
| 31 | \( 1 - 32.2T + 961T^{2} \) |
| 37 | \( 1 + (21.0 + 21.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 51.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (24.7 - 24.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-52.0 - 52.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-26.3 + 26.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 103. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 119.T + 3.72e3T^{2} \) |
| 67 | \( 1 + (49.4 + 49.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 94.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (15.0 - 15.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 112. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-13.6 + 13.6i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 35.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-72.1 - 72.1i)T + 9.40e3iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.521754337583028248076327336183, −8.839278623375742839828357328592, −8.000731608411832450068763088812, −6.89660011417841348408195555750, −6.28146252638904057514149216528, −5.24092547668618733941903728419, −4.65593600382842295746945068974, −3.57442361750090325407184834060, −2.45921159479088747467780013332, −0.52451817926683412994597728047,
1.32767285322409793608043100144, 2.06285941273499148069622346266, 3.78106959451477965404017365166, 4.23879929898973536260170458930, 5.51063250548459024841335715790, 6.38132546080134268718896260716, 6.88023167275282815146722644300, 8.356414173836533837259407943561, 8.730549834090956157350777943554, 10.23452958774463304696630458904